For a fiber-reinforced composite, the efficiency of reinforcement η is dependent on fiber length l according to where x represents the length of the fiber at each end that does not contribute to the load transfer. What length is required for a 0.62 efficiency of reinforcement, assuming that x is 0.61 mm?

Answer:

The minimum effective focal length of the focusing mechanism (lens plus cornea) of the typical eye is 2.27 cm.

Explanation:

The diameter of person's eye is 2.5 cm. The close point or the near point of the eye is 25 cm and the far point is infinity. We need to determine the minimum effective focal length of the focusing mechanism (lens plus cornea) of the typical eye. Let f is the minimum effective focal length. It can be calculated using lens formula as :

[tex]\dfrac{1}{f}=\dfrac{1}{25}+\dfrac{1}{2.5}[/tex]

f = 2.27 cm

So, the minimum effective focal length of the focusing mechanism (lens plus cornea) of the typical eye is 2.27 cm and it is at the nearest point. Hence, this is the required solution.

Answer:

The ice is cooler than the table.

Explanation:

The difference in temperature is a necessary condition for the transfer of heat between two bodies.

Here if the heat is being transferred from the table to an ice block this means that the temperature of the table is greater than the ice which drives the heat energy from the table to the ice block and the heat will continue to flow until they both attain a common temperature.

However,

Final answer:

Heat flow from a table to a block of ice enables the ice to undergo a phase change from solid to liquid, known as the latent heat of fusion, without a change in temperature .

Explanation:

If heat is flowing from a table to a block of ice moving across the table, which of the following must be true? The correct answer from the choices provided is that the ice is changing phase. Heat flow from the table to the ice suggests that the ice is absorbing energy. When ice absorbs heat, it undergoes a phase change, where the ice melts and becomes liquid water without any change in temperature. This process is known as the latent heat of fusion, where heat is transferred to cause a phase change without altering the temperature of the system. Therefore, while the table may be rough and friction may exist, and the ice might indeed be cooler than the table, the guaranteed phenomenon occurring is a phase change from solid to liquid as the ice absorbs heat.

Answer:

c

Explanation:

from. 50 to 60 diopters

The range of Anna's vision will be between 0-5 diopters.

It is given that

Anna is able to focus on objects within 20 cm of her eyes (her near point)

Now the distance between the cornea and retina, of Anna's eye is 20 mm

The focal length will be f= 20+2= 22cm

Now the power of eyes will be given by

[tex]P= \dfrac{1000}{f(in \ mm)}[/tex]

[tex]P= \dfrac{1000}{ 22}=4.54 \ diopters[/tex]

Thus the range of Anna's vision will be between 0-5 diopters.

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Answer:

D) Rotate the loop about an axis that is directed in the z direction and that passes through the center of the loop

Explanation:

The magnetic flux is defined as the total magnetic field times the area normal to the magnetic field lines.

Mathematically:

[tex]\phi_B=\vec{B}.\vec{A}[/tex]

where:

[tex]\vec{A}=[/tex] area vector directed normal to the surface

[tex]\vec{B}=[/tex] magnetic field vector

Final answer:

Rotating the conducting loop about an axis in the z direction will not change the magnetic flux through the loop because the angle between the magnetic field and the normal to the plane of the loop remains constant. so the correct option is D

Explanation:

The question pertains to the change in magnetic flux through a conducting loop when subjected to different actions. According to Faraday's Law of Electromagnetic Induction, the magnetic flux through a loop is given by the product of the magnetic field strength, the area of the loop, and the cosine of the angle between the magnetic field direction and the normal to the loop. In options A, B, and C, changes to the area or the strength of the magnetic field alter the magnetic flux since these factors directly affect the calculation of flux. Option D involves rotating the loop about an axis in the z direction; such a rotation does not change the angle between the magnetic field and the normal to the loop's plane, therefore would not change the flux. In contrast, option E, where the loop is rotated about an axis in the y direction, changes this angle and thus the flux.

Therefore, the action that will not change the magnetic flux through the loop is:

D) Rotate the loop about an axis that is directed in the z direction and that passes through the center of the loop.

Complete Question:

A machinist turns the power on to a grinding wheel, which is at rest at time t = 0.00 s. The wheel accelerates uniformly for 10 s and reaches the operating angular velocity of 25 rad/s. The wheel is run at that angular velocity for 37 s and then power is shut off. The wheel decelerates uniformly at 1.5 rads/s2 until the wheel stops. In this situation, the time interval of angular deceleration (slowing down) is closest to

Answer:

t= 16.7 sec.

Explanation:

As we are told that the wheel is accelerating uniformly, we can apply the definition of angular acceleration to its value:

γ = (ωf -ω₀) / t

If the wheel was at rest at t-= 0.00 s, the angular acceleration is given by the following equation:

γ = ωf / t = 25 rad/sec / 10 sec = 2.5 rad/sec².

When the power is shut off, as the deceleration is uniform, we can apply the same equation as above, with ωf = 0, and ω₀ = 25 rad/sec, and γ = -1.5 rad/sec, as follows:

γ= (ωf-ω₀) /Δt⇒Δt = (0-25 rad/sec) / (-1.5 rad/sec²) = 16.7 sec

Answer:

a. [tex]m=\frac{2f_{s}tan\theta}{g}[/tex]

b. m=5.99kg

Explanation:

a.

In order to solve this problem, we can start by drawing a diagram of the situation. Drawing a diagram is really important since it will help use understand the problem better and analyze it as well. (See attached picture).

In the diagram we can see the forces that are acting on the ladder. We will assume the ladder is static (this is it doesn't have any  movement) and analyze the respective forces in the x-direction and the forces in the y-direction, as well as the moments about point B.

So we start with the sum of forces about y, so we get:

[tex]\sum F_{y}=0[/tex]

N-W=0

N=W

N=mg

Next we can do the sum of forces about x, so we get:

[tex]\sum F_{x}=0[/tex]

which yields:

[tex]f_{s}-F_{W}=0[/tex]

so:

[tex]f_{s}=F_{W}[/tex]

Next the torque about point B, so we get:

[tex]\sum M_{B}=0[/tex]

so:

[tex]f_{s} L sin\theta - NLcos\theta + \frac{WL}{2}cos\theta = 0[/tex]

From the sum of forces in the y-direction we know that N=mg (this is because the wall makes no friction over the ladder) so we can directly substitute that into our equation, so we get:

[tex]f_{s} L sin\theta - WLcos\theta + \frac{WL}{2}cos\theta = 0[/tex]

We can now combine like terms, so we get:

[tex]f_{s} L sin\theta -\frac{WL}{2}cos\theta = 0[/tex]

we know that W=mg, so we can substitute that into our equation, so we get:

[tex]f_{s} L sin\theta -\frac{mgL}{2}cos\theta = 0[/tex]

which can now be solved for the mass m:

[tex]\frac{mgL}{2}cos\theta = f_{s} L sin\theta[/tex]

If we divided both sides of the equation into L, we can see that the L's get cancelled, so our equation simplifies to:

[tex]\frac{mg}{2}cos\theta = f_{s}sin\theta[/tex]

we can now divide both sides of the equation into g so we get:

[tex]\frac{m}{2}cos\theta = \frac{f_{s}sin\theta}{g}[/tex]

next we can divide both sides of the equation into cos θ so we get:

[tex]\frac{m}{2}= \frac{f_{s}sin\theta}{g cos\theta}[/tex]

and finally we can multiply both sides of the equation by 2 so we get:

[tex]m=\frac{2f_{s}sin\theta}{g cos\theta}[/tex]

we know that:

[tex]tan \theta=\frac{sin \theta}{cos \theta}[/tex]

so we can simplify the equation a little more, so we get:

[tex]m=\frac{2f_{s}tan \theta}{g}[/tex]

b.

So now we can directly use the equation to find the mass of the ladder with the data indicated by the problem:

θ=32° and [tex]f_{s}=47N[/tex]

we also know that [tex]g=9.8m/s^{2}[/tex]

so we can use our equation now:

[tex]m=\frac{2f_{s}tan \theta}{g}[/tex]

so we get:

[tex]m=\frac{2(47N)tan (32^{o})}{9.8m/s^{2}}[/tex]

which yields:

m=5.99kg

Answer:

a) m = 2fs(tanθ)/g

b) m = 5.99 kg

Explanation:

a) Expression for the ladder's mass, m, in terms of θ, g, fs, and/or L.

Data

m₁ : mass of the lader

g: acceleration due to gravity

L : ladder length

θ  : angle that makes  the  ladder  with the floor

µ = 0 : coefficient of friction between the ladder and the wall

fs : friction force between the ladder and the floor

Forces acting on the ladder

W =m*g : Weight of the ladder (vertical downward) , m: mass of the lader 

FN :Normal force that the floor exerts on the ladder (vertical upward) (point A)

fs : friction force that the floor exerts on the ladder (horizontal to the left) (point A)

N : Forces that the wall exerts on the ladder (horizontal to the right)

Equilibrium  of the forces in X

∑Fx=0

N -fs = 0

N = fs

The equilibrium equation of the moments at the point contact point of the ladder with the floor:

∑MA = 0  

MA = F*d  

Where:  

∑MA : Algebraic sum of moments in the the point (A) (contact point of the ladder with the wall)  

MA : moment in the point A ( N*m)  

F : Force ( N)  

d :Perpendicular distance of the force to the point A ( m )

Calculation of the distances of the forces at the point A

d₁ = (L/2)*cosθ : Distance from W to the point A

d₂ = L*sinθ : Distance from N to the point A

Equilibrium of the moments at the point A

∑MA = 0  

N(d₂)-W(d₁) = 0

W( (L/2)*cosθ)= N(L*sinθ )

mg( (L/2)*cosθ)= fs(L*sinθ )

We divided by L both sides of the equation

mg (cosθ/2) =fs(sinθ)

m=2fs(sinθ)/ g( cosθ)

m = 2fs(tanθ)/ g

b) If θ=32° and the friction force on the floor is 47 N, what is the mass of the ladder?

m = 2fs(tanθ)/ g

m = 2(47)(tan32°)/(9,8)

m = 5.99 kg

Answer:C

Explanation:

There are two types of cells in the eyes rod and cone cells. Rod cells Provide vision during the night or dim light also called of scotopic vision whereas cone cells provide vision during day time or at bright light also called photopic vision.  

Rod cells do not support the color vision that is why it is difficult to differentiate between colors.

Also, the amount of light entering the eyes is low as result cone cells are unable to stimulate.

It is hard to see color at night because cones, which are responsible for color vision, need more light to be stimulated than rods, which allow us to see in low light but only in grayscale.

The best explanation for why it is difficult to discriminate the color of an object at night is option C. At night, the amount of light entering the eye is insufficient to stimulate the cone cells but is sufficient to stimulate the rod cells. The rods are highly sensitive to light, allowing us to see in low light conditions but do not provide color information. Cones require brighter light to function and are responsible for our color vision. Since cones do not react to low-intensity light, our vision at night is predominantly in shades of gray, serviced by the activity of rods.

The sets of driver's actions “Signalling; Changing Gears; Checking mirrors” are multi-task performances necessary for the safe vehicle operation.

Answer: Option A

Explanation:

Signaling is what drivers plan to do is important for security because other drivers can only know if he tells them. Here are some general rules about Signalling or turns.

The correct gear change is important. If he can't put the vehicle in the right gear while driving, he has less control. And also, he must check the exterior mirrors on both sides regularly. He needs to check his vehicle mirrors to make sure nobody stands or walks past him.

Answer:

The difference between frictionless ramp and a regular ramp is that on a frictionless ramp the ball cannot roll it can only slide, but on a regular ramp the ball can roll without slipping.

We will use conversation of energy.

[tex]K_A_1 + U_A_1 = K_A_2 + U_A_2\\\frac{1}{2}I\omega^2 + \frac{1}{2}mv^2 + 0 = 0 + mgH_A[/tex]

Note that initial potential energy is zero because the ball is on the bottom, and the final kinetic energy is zero because the ball reaches its maximum vertical distance and stops.

For the ball B;

[tex]K_B_1 + U_B_1 = K_B_2 + U_B_2[/tex]

[tex]\frac{1}{2}I_B\omega^2 + \frac{1}{2}mv^2 + 0 = 0 + mgH_B[/tex]

The initial velocities of the balls are equal. Their maximum climbing point will be proportional to their final potential energy. Since their initial kinetic energies are equal, their final potential energies must be equal as well.

Hence, both balls climb the same point.

Explanation:

Both balls A and B will climb up to the same height according to the conservation of energy.

Since the balls are identical, let both have mass m and velocity v.

Also the moment of inertia I and the angular speed ω will be the same.

Ball A encounters a frictionless ramp:

On a frictionless ramp, the ball slides down the ramp since it cannot roll as there is no friction present. Since there is o frictional force, there is no dissipation of energy. Energy is conserved.

According to the law of conservation of energy, the total energy of the system must be conserved.

KE(initial) + PE(initial) = KE(final) + PE(final)

[tex]\frac{1}{2}I\omega^2+\frac{1}{2}mv^2+0=0+mgH_A\\\\H_A=\frac{1}{mg} [\frac{1}{2}I\omega^2+\frac{1}{2}mv^2][/tex]

initial KE has rotational and translational kinetic energy and the initial PE is zero since the ball is on the ground, also the final KE is zero since the velocity at the highest point will be zero.

Ball B encounters a regular ramp:

On a regular ramp, the ball can roll without sliding due to friction.

[tex]\frac{1}{2}I\omega^2+\frac{1}{2}mv^2+0=0+mgH_B\\\\H_B=\frac{1}{mg} [\frac{1}{2}I\omega^2+\frac{1}{2}mv^2][/tex]

The height will be the same since the velocity is the same as ball A.

Also, the translational or rotational velocity will be zero at the highest point.

Hence, both balls climb the same height.

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Answer:

Induced emf in the coil, [tex]\epsilon=4.05\ volts[/tex]

Explanation:

Given that.

Number of turns in the coil, N = 200

Side of square, d = 18 cm = 0.18 m

The field changes linearly from 0 to 0.50 T in 0.80 s.

To find,

The magnitude of the induced emf in the coil while the field is changing.

Solution,

We know that due to change in the magnetic field, an emf gets induced in the coil. The formula of induced emf is given by :

[tex]\epsilon=\dfrac{d\phi}{dt}[/tex]

[tex]\phi[/tex] = magnetic flux

[tex]\epsilon=-\dfrac{d(NBA)}{dt}[/tex]

A is the ares of square

[tex]\epsilon=AN\dfrac{d(B)}{dt}[/tex]

[tex]\epsilon=AN\dfrac{B_f-B_i}{t}[/tex]

[tex]\epsilon=(0.18)^2\times 200 \times \dfrac{0.5-0}{0.8}[/tex]

[tex]\epsilon=4.05\ volts[/tex]

So, the induced emf in the coil is 4.05 volts. Hence, this is the required solution.

To calculate the induced emf in a 200-turn coil with a changing magnetic field, use Faraday's Law of Induction. The magnitude of the induced emf is 4.05 V. This result is obtained through calculations involving the number of turns, area of each turn, and the rate of change of the magnetic field.

The problem involves calculating the induced emf in a coil with 200 turns of wire, where each turn is a square of side d = 18 cm (or 0.18 m). The magnetic field (B) changes linearly from 0 to 0.50 T over a time interval of 0.80 s.

We use Faraday's Law of Induction, which states that the induced emf (ε) in a coil is given by:

ε = -N(dΦ/dt)

where:

The magnetic flux (Φ) through one turn is calculated as:

Φ = B × A

where A is the area of one turn of the coil:

A = d × d = 0.18 m × 0.18 m = 0.0324 m²

Since the magnetic field changes linearly, the rate of change of the magnetic field (dB/dt) is:

dB/dt = (0.50 T - 0 T) / 0.80 s = 0.625 T/s

Now, we can calculate the rate of change of the magnetic flux (dΦ/dt):

dΦ/dt = A × dB/dt = 0.0324 m² × 0.625 T/s = 0.02025 Wb/s

Finally, we use Faraday's Law to find the induced emf:

ε = -N(dΦ/dt) = -200 (0.02025 Wb/s) = -4.05 V

The magnitude of the induced emf in the coil while the field is changing is 4.05 V.

Answer:

a. get warmer.

Explanation:

When the water vaporous reach the upper layer of the atmosphere they get a cooler air to which they loose their temperature and condense to form clouds as a the temperature of the air increases.

It may be noted that the water looses its high amount of latent heat of vaporization to condense into water this significantly increases the temperature of the air in contact.

The smallest possible value for the total energy of an electron with l=5 in a hydrogen atom is -0.37778 eV.

The smallest possible value for the total energy of an electron in a hydrogen atom can be found by using the energy formula for Bohr's model. The energy is given by:

E = -13.6 eV / n^2

where n is the principal quantum number. In this case, since l=5, the smallest possible value for the total energy is when n=6. Plugging this value into the energy formula, we get:

E = -13.6 eV / 6^2 = -0.37778 eV

Answer:

[tex]1.04\times 10^7\ J.[/tex]

Explanation:

In the question given :

Pressure is constant

Therefore, Work done, [tex]W=P\times\Delta V[/tex]

Pressure, P=1.01 × 105 Pa.

Final volume, [tex]V_f=8.50\ m^3.[/tex]

Initial volume, [tex]V_i=\dfrac{Mass}{density}=\dfrac{5}{958}=5.22\times10^-3\ m^3.[/tex]

Therefore, W=8.58\times 10^{5}\ J.

Also, Heat Given, [tex]Q=m\times L=5\times 2.26\times 10^{6}\ J=1.13\times 10^7\ J.[/tex]

Also, according to First law of thermodynamics:

[tex]\Delta U=Q-W=(1.13\times 10^7)-(8.58\times 10^5)=1.04\times 10^7\ J.[/tex]

Hence, this is the required solution.

The energy needed to convert all of the water into steam in this isothermal system is 11.30 MJ. This is calculated using the formula Q = mLv, substituting the given values for mass and latent heat of vaporization.

The heat required to convert water from liquid into steam, in an isothermal process, can be found using the formula Q = mLv, where 'm' is the mass of the water, and 'Lv' is the latent heat of vaporization.

Given that we have 5.00 kg of water and the heat of vaporization for water under atmospheric pressure is 2.26 × 106 J/kg, we can substitute these values into our formula:

Q = 5.00 kg × 2.26 × 106 J/kg.

This gives us Q = 11.30 × 106 J, or 11.30 MJ. Therefore, 11.30 MJ of heat is added to the system to convert all the water into steam at atmospheric pressure.

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Final answer:

The magnitude of the electric field just outside the surface of a sphere designed to deflect solar radiation on the moon or Mars is 10⁷ N/C, calculated based on the given requirements for the electric field at a distance from the sphere.

Explanation:

The question asks for the magnitude of an electric field E just outside the surface of a sphere that is proposed to protect astronauts on the surface of the moon or Mars by deflecting high-energy charged particles emitted by the sun.

The sphere, assumed to have a uniform surface charge distribution, generates an electric field whose magnitude at a distance r from its center can be found using the formula E = kQ/r², where k is Coulomb's constant (8.99 x 10⁹ N·m²/C²), Q is the charge on the sphere, and r is the distance from the center of the sphere.

Given that the sphere's electric field's magnitude at a distance of 25 m from its center needs to be 1 x 10⁶ N/C, we find the charge Q required to produce this field.

Once Q is determined, we can calculate the electric field's magnitude just outside the sphere's surface (at r = 2.5 m, which is the radius of the 5 m diameter sphere) using the same formula.

The calculation reveals that the magnitude of E just outside the surface of the sphere is of the order 10⁷ N/C, making option c) 10⁷ N/C the correct answer.

The magnitude of the electric field just outside the surface of the sphere is [tex]\(10^7 \, \text{N/C}\),[/tex] so the correct answer is (c) [tex]\(10^7 \, \text{N/C}\).[/tex]

To find the magnitude of the electric field just outside the surface of the sphere, we'll use Gauss's Law, which states:

[tex]\[\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}\][/tex]

Where:

- \(\vec{E}\) is the electric field vector

- \(d\vec{A}\) is a differential area vector

- \(Q_{\text{enc}}\) is the enclosed charge

- \(\varepsilon_0\) is the permittivity of free space

Since the electric field is radial, we can express it as [tex]\(E = E(r)\hat{r}\)[/tex], where [tex]\(\hat{r}\)[/tex] is a unit vector pointing radially outward from the center of the sphere.

The magnitude of the electric field \(E\) just outside the surface of the sphere is equal to the magnitude of the electric field produced by a point charge at the center of the sphere, which is given by Coulomb's Law:

[tex]\[E = \frac{k |Q|}{r^2}\][/tex]

Where:

- \(k\) is Coulomb's constant [tex](\(8.99 \times 10^9 \, \text{N}\cdot\text{m}^2/\text{C}^2\))[/tex]

- \(|Q|\) is the magnitude of the charge enclosed by the Gaussian surface

- \(r\) is the distance from the center of the sphere

Given that [tex]\(E = 1 \times 10^6 \, \text{N/C}\) at \(r = 25 \, \text{m}\), we can solve for \(|Q|\):[/tex]

[tex]\[1 \times 10^6 \, \text{N/C} = \frac{k |Q|}{(25 \, \text{m})^2}\][/tex]

[tex]\[|Q| = \frac{(1 \times 10^6 \, \text{N/C}) \cdot (25 \, \text{m})^2}{k}\][/tex]

[tex]\[|Q| = \frac{(1 \times 10^6 \, \text{N/C}) \cdot (625 \, \text{m}^2)}{8.99 \times 10^9 \, \text{N}\cdot\text{m}^2/\text{C}^2}\][/tex]

[tex]\[|Q| = \frac{625 \times 10^6}{8.99}\][/tex]

[tex]\[|Q| \approx 69.633 \, \text{C}\][/tex]

Now, let's calculate the electric field just outside the surface of the sphere using this charge:

[tex]\[E = \frac{k |Q|}{r^2}\][/tex]

[tex]\[E = \frac{(8.99 \times 10^9 \, \text{N}\cdot\text{m}^2/\text{C}^2) \cdot (69.633 \, \text{C})}{(5 \, \text{m})^2}\][/tex]

[tex]\[E = \frac{(8.99 \times 10^9 \, \text{N}\cdot\text{m}^2/\text{C}^2) \cdot (69.633 \, \text{C})}{25 \, \text{m}^2}\][/tex]

[tex]\[E = \frac{629.586 \times 10^9}{25}\][/tex]

[tex]\[E \approx 25.183 \times 10^7 \, \text{N/C}\][/tex]

Therefore, the magnitude of the electric field just outside the surface of the sphere is approximately [tex]\(2.5183 \times 10^8 \, \text{N/C}\).[/tex]

The closest option is d) [tex]\(10^8 \, \text{N/C}\).[/tex]

Answer:

the demonstrations are in the description.

Explanation:

First of all, reference is made to a picture that may be the one shown attached.

when the string hits the pin, the sphere will continue to swing to the right until a height h'.

a) Initially, the system has only potential energy because it is released with zero velocity from a height h.

if we consider that the system has no energy losses, then all that potential energy must be conserved. Thus, this energy is transformed in principle into kinetic energy and once the kinetic energy is maximum, it will be transformed again into potential energy.

The potential energy is given by:

[tex]E_p = mgh[/tex]

As we can see, potential energy only depends on mass, height and gravity. Since mass and gravity are constant, then for potential energy to be conserved, heights must be the same.

b) the statement probably has a transcription error in question b. Corrected, it would look like:

Show that if the pendulum is released from the horizontal position (∡ = 90) and is to swing in a complete circle centered on the peg, the minimum value of d must be 3L/5.

podemos asumir que la altura cero está dada en el punto de reposo del péndulo, por lo cual la energía inicial (que es potencial) está dada por:

[tex]E_0=MGL[/tex]

For the pendulum to swing, the tension must always be positive, and as a consequence, the centripetal force must be greater or at least equal to the weight. Mathematically

[tex]mV^2 /(L-d) = mg[/tex]  (1)

Where:

[tex]mV^2 /(L-d) [/tex] It is the centripetal force that the sphere experiences.

Now, for the sphere to describe the circular motion, the minimum kinetic energy must be:

[tex]\frac{mV^2}{2} = mgL-2mg(L-d)[/tex]   (2)

Then, replacing equation 1 in 2:

[tex]\frac{mg(L-d)}{2} = mgL-2mg(L-d)[/tex]

Simplifying:

[tex]L-2L+2d= \frac{L}{2} - \frac{d}{2}[/tex]

[tex]\frac{5}{2}d= \frac{3}{2}L[/tex]

Which means

[tex]d=\frac{3L}{5}[/tex]

Final answer:

When a pendulum hits a peg, it will return to its original height if released from below the peg. The minimum value of d, in order for a pendulum to swing in a complete circle centered on a peg, must be 3L/5.

Explanation:

A pendulum comprising a light string and a small sphere swings in the vertical plane. When the string hits a peg located below the point of suspension, the sphere will return to its original height if it is released from a height below that of the peg. The reason for this is that when the string hits the peg, the tension in the string abruptly changes direction, causing the sphere to swing back upwards.

If the pendulum is released from rest at the horizontal position and is to swing in a complete circle centered on the peg, the minimum value of d must be 3L/5. This is because when the pendulum is at the lowest point, the tension in the string must be equal to or greater than the weight of the sphere in order to maintain circular motion. The minimum value of d ensures that this condition is satisfied.

Answer:

Explanation:

position of centre of mass of door from surface of water

= 10 + 1.1 / 2

= 10.55 m

Pressure on centre of mass

atmospheric pressure + pressure due to water column

10 ⁵ + hdg

= 10⁵ + 10.55 x 1000 x 9.8

= 2.0339 x 10⁵ Pa

the net force acting on the door (normal to its surface)

= pressure at the centre x area of the door

= .9 x 1.1 x 2.0339 x 10⁵

= 2.01356 x 10⁵ N

pressure centre will be at 10.55 m below the surface.

When the car is filled with air or  it is filled with water , in both the cases pressure centre will lie at the centre of the car .

The net force acting on the door (normal to its surface) and the location of the pressure should be  [tex]2.01356 \times 10^5 N[/tex]

The position of center of mass of door from the surface of water should be

[tex]= 10 + 1.1 \div 2[/tex]

= 10.55 m

Now

Pressure on center of mass

= atmospheric pressure + pressure due to water column

[tex]= 10^5 + 10.55 \times 1000 \times 9.8\\\\= 2.0339 \times 10^5 Pa[/tex]

Now

the net force acting on the door (normal to its surface)

[tex]= .9 \times 1.1 \times 2.0339 \times 10^5\\\\= 2.01356 \times 10^5 N[/tex]

So,

pressure centre will be at 10.55 m below the surface.

Therefore,

When the car is filled with air or  it is filled with water , in both the cases pressure center should be lie at the centre of the car .

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Answer:

a) dh/dt = -44.56*10⁻⁴ cm/s

b) dr/dt = -17.82*10⁻⁴ cm/s

Explanation:

Given:

Q = dV/dt = -35 cm³/s

R = 1.00 m

H = 2.50 m

if h = 125 cm

a) dh/dt = ?

b) dr/dt = ?

We know that

V = π*r²*h/3

and

tan ∅ = H/R = 2.5m / 1m = 2.5  ⇒ h/r = 2.5

⇒  h = (5/2)*r

⇒  r = (2/5)*h

If we apply

Q = dV/dt = -35 = d(π*r²*h/3)*dt

⇒  d(r²*h)/dt = 3*35/π = 105/π   ⇒   d(r²*h)/dt = -105/π

a) if   r = (2/5)*h

⇒  d(r²*h)/dt = d(((2/5)*h)²*h)/dt = (4/25)*d(h³)/dt = -105/π

⇒  (4/25)(3*h²)(dh/dt) = -105/π

⇒  dh/dt = -875/(4π*h²)

b) if  h = (5/2)*r

Q = dV/dt = -35 = d(π*r²*h/3)*dt

⇒  d(r²*h)/dt = d(r²*(5/2)*r)/dt = (5/2)*d(r³)/dt = -105/π

⇒  (5/2)*(3*r²)(dr/dt) = -105/π

⇒  dr/dt = -14/(π*r²)

Now, using h = 125 cm

dh/dt = -875/(4π*h²) = -875/(4π*(125)²)

⇒  dh/dt = -44.56*10⁻⁴ cm/s

then

h = 125 cm  ⇒  r = (2/5)*h = (2/5)*(125 cm)

⇒  r = 50 cm

⇒  dr/dt = -14/(π*r²) = - 14/(π*(50)²)

⇒  dr/dt = -17.82*10⁻⁴ cm/s

The rate at which the depth of the water is changing is -0.178 cm/sec, and the rate at which the radius of the top of the water is changing is -0.14 cm/sec.

The subject of this question is related to Calculus and the specific topic is related rates. For this problem, in addition to the rate at which water is leaking, we need to consider the geometric relationship within the cone-shaped water tank.

First, we are given that V' = -35 cm^3/sec (we make it negative because the volume is decreasing) and we know that the volume of a cone is V = (1/3)πr²h. Our tank parameters are r = 1m and h = 2.5m, but we need everything in the same units, so we convert our radius to 100 cm.

We know through similar triangles that r/h = R/H, where r and R are the radii at any given time, and h and H are the heights at any given time respectively. Thus, r = (Rh)/H. Substituting r in our volume equation, we get: V = (1/3)π((Rh)^2)/H=h²πR²/H, and hence V=hπR². Differentiating this implicitly with t gives V' = h'πR²+2hπRr'.

Substituting for V', h and r from our given information, we get: -35=h'πR²+2(1.25)πRr'. We can solve for h' and r' separately.

(a) To find h' we set r' = 0, because we only want to know how depth h is changing. Solving for h' we find it to be -0.178 cm/sec.

(b) To find r', we set h' = 0, because we are only interested in how radius r is changing. Solving for r' we get -0.14 cm/sec.

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To solve this problem we will use the concept of the Doppler effect applied to the speed of blood, the speed of sound in the blood and the original frequency. This relationship will also be extrapolated to the frequency given by the detector and measured the change in frequencies through the beat frequency. So:

[tex]f_{blood} = f (1-\frac{v_{blood}}{v_{snd}})[/tex]

Where

[tex]f_{blood}[/tex] = Frequency of the blood flow

f = Frequency of the original signal

[tex]v_{blood}[/tex] = Speed of the blood flow

[tex]v_{snd}[/tex] = Speed of sound in blood

[tex]f''_{detector} = \frac{f_{blood}}{(1+\frac{v_{blood}}{v_{snd}})}[/tex]

[tex]f''_{detector} = f (\frac{(1-\frac{v_{blood}}{v_{snd}})}{(1+\frac{v_{blood}}{v_{snd}})})[/tex]

[tex]f''_{detector} = f \frac{(v_{snd}-v_{blood})}{(v_{snd}+v_{blood})}[/tex]

Now calculating the beat frequency is

[tex]\Delta f = f-f''_{detector}[/tex]

Replacing this latest value we have that,

[tex]\Delta f = f-f \frac{(v_{snd}-v_{blood})}{(v_{snd}+v_{blood})}[/tex]

[tex]\Delta f = f \frac{2v_{blood}}{v_{snd}+v_{blood}}[/tex]

Replacing we have,

[tex]\Delta f = (3.5*10^6)(\frac{2*(3*10^{-2})}{1.54*10^3+3*10^{-2}})[/tex]

[tex]\Delta f = 136.36Hz[/tex]

Therefore the beat frequency is 136.36Hz

Using the beat frequency relation, the expected beat frequency observed at the flow meter would be 136.36 Hz

Given the Parameters :

The expected beat frequency observed can be calculated uisng the relation :

Substituting the values into the formula :

[tex] \delta F = 3.5 \times 10^{3} \frac{2 \times 0.03}{(1540 + 0.03}[/tex]

[tex] \delta F = 3.5 \times 10^{3} \frac{0.06}{(1540.03}[/tex]

[tex] \delta F = 136.36 Hz [/tex]

Therefore, the expected beat frequency observed at the flow meter will be 136.36 Hz

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Answer:

[tex]|\vec{F}| =48.60\ N[/tex]

Explanation:

given,

Current in the power line = I = 5500 A

earth's magnetic field = 60.0 μT

inclination downward = 67°

Length = 160 m

magnetic force = ?

[tex]\vec{F} = I (\vec{L}\times \vec{B})[/tex]

[tex]|\vec{F}| = I |(\vec{L}\times \vec{B})|[/tex]

[tex]|\vec{F}| =I LB sin \theta [/tex]

[tex]|\vec{F}| = 5500 \times 160 \times 60 \times 10^{-6}\times sin 67^0[/tex]

[tex]|\vec{F}| =48.60\ N[/tex]

According to the right hand rule the direction of the force is perpendicular to the plane of the length and the magnetic field so, it is to west.

Answer:

Energy required = [tex]J=2.73\times10^7 \ J.[/tex]

Explanation:

We know according to zeroth law of Thermodynamics, all bodies in a close system remains in thermal equilibrium.

Means, Energy gain by one part of system = energy loss by one another part of system.

Here, energy is given to surrounding and it is lost by water.

Therefore, its temperature decreases.

Energy required to convert 82 kg liquid, E = [tex]m\times L[/tex].

Here, [tex]m=82 \ kg[/tex]

         [tex]L=larent \ heat \ of \ fusion=3.34\times 10^5J/kg.[/tex].

E=[tex]82 \times 3.34\times 10^5\ J=2.73\times10^7 \ J.[/tex]

Answer:

U. With no variation.

Explanation:

Note- since temperature remains constant when pressure becomes twice and volume becomes half, and internal energy of ideal gas is function of only temperature so it remains constant. The internal energy is independent of the variables stated in the exercise.

Answer:

Correct answer is (b) Revitalization movements always fail because they require too much change to be tolerated.

Example of a revitalization movement is the Ghost Dance that swept through western Native American cultures from 1870-1890.

Explanation:

Another example of Revitalization movements are mostly associated with religion. They often occur in disorganized societies due to warfare, revolution but not necessarily an animistic societies.

The incorrect statement is that 'Revitalization movements always fail because they require too much change to be tolerated.' Revitalization movements have led to significant societal changes and have even given rise to major religions like Christianity, Judaism, and Islam. The correct option is b.

The statement 'Revitalization movements always fail because they require too much change to be tolerated' is incorrect. Revitalization movements are deliberate, organized, conscious efforts by members of a society to construct a more satisfying culture. While it's true that they can face intense resistance because they often push for significant changes, it's not true that they always fail. Many have had long-lasting impacts. For example, the Protestant Reformation was a revitalization movement that sought to reform the Roman Catholic Church and led to the creation of Protestant churches. It instigated enormous change in society, and it certainly didn't fail.

All major religions, including Judaism, Christianity, and Islam, did indeed begin as revitalization movements. These movements were adapted to the changing needs of society, and they have resulted in lasting religions that are still prevalent today. Hence revitalization movements can have profound and lasting impacts on society, institutions and cultures, far from always being impractical or bound to fail.

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Answer:

If the period of the wave is increased by the factor of 2.70, the wavelength of the wave is also increased by a factor of 2.70. So,

[tex]\lambda_2 = 235\times 2.70 = 634.5 ~nm[/tex]

The magnetic field component can be written as

[tex]\vec{B} = \frac{E_{max}}{c}e^{i(\vec{k}\vec{z}-\omega t)}\^{z}[/tex]

The magnetic field is in the z-direction, because the E-field is directed towards +y and the wave is propagating in the +x-direction. The right-hand rule gives us the direction of the B-field.

[tex]\vec{E} \times \vec{B} = \vec{S}[/tex]

S is the Poynting vector which gives us the propagation of the wave.

We will use the following relationships

[tex]k = 2\pi / \lambda\\f = \omega / 2\pi\\c = \lambda f = \lambda \omega / 2\pi\\\omega = 2\pi c/\lambda[/tex]

[tex]\vec{B} = \frac{7.7\times 10^{-3}}{3\times 10^8}e^{(\frac{2\pi}{3\times 10^8}z - \frac{2\pi\times 3\times 10^8}{634.5})}\\\vec{B} = 2.56\times10^{-11} e^{(2.09\times10^{-8}z - 2.96\times10^{6}t)}\^{z}[/tex]

To find the resulting magnetic field equation of an electromagnetic wave after the period is increased, we use relationships between the wave's wavelength, frequency, period, and electric and magnetic fields, applying the changes to the wave's properties based on Maxwell's theory of electromagnetism.

The question is asking for the equation of the resulting magnetic field component of an electromagnetic wave after its period has been increased by a factor of 2.70. Given the initial wavelength λ of 235 μm (micrometers) and the maximum electric field Emax in the +y direction of 7.70×10⁻³ V/m, we can find the initial frequency ν using the relationship ν = c / λ, where c is the speed of light. Since the period T is the reciprocal of the frequency (ν = 1/T), when the period is increased by a factor of 2.70, the frequency decreases by the same factor. The maximum magnetic field Bmax is related to the maximum electric field Emax by Bmax = Emax / c. The resulting magnetic field B can be described by the function B(x, t) = Bmax sin(kx - ωt + φ), where k is the wave number, ω is the angular frequency (2πν), and φ is the phase constant.

Answer:

0.1587

Explanation:

Given data

μ = 0.5 mg  

standard deviation σ =0.1 mg

n=100

we know that

[tex]P(\overline X<0.49)[/tex]

[tex]Z= (\frac{\overline X-\mu}{\sigma/\sqrt{n} })[/tex]

putting the values we get

[tex]Z= (\frac{0.49-0.5}{0.1/\sqrt{100} })[/tex]

Z=-1

Area under the curve for z =-1 is 0.1587 (from z score table)

P(X<0.49) = 0.1587

P(X<0.37) = 0.0013

Answer:

-0.23694 N

Explanation:

g = Acceleration due to gravity = 9.81 m/s²

m = Mass of the Earth =  5.972 × 10²⁴ kg

G = Gravitational constant = 6.67 × 10⁻¹¹ m³/kgs²

r = Radius of Earth = 6371000 m

dr = Height = 1 mile = 1609.34 m

Acceleration is given by

[tex]a=\dfrac{GM}{r^2}[/tex]

Change in acceleration is given by

[tex]da=-2\dfrac{GM}{r^3}dr[/tex]

[tex]w=ma\\\Rightarrow w=m\dfrac{GM}{r^2}\\\Rightarrow w=469\ N[/tex]

[tex]dw=mda\\\Rightarrow dw=-m2\dfrac{GM}{r^3}dr\\\Rightarrow dw=-2w\dfrac{dr}{r}\\\Rightarrow dw=-2\times 469\times \dfrac{1609.34}{6.371\times 10^{6}}\\\Rightarrow dw=-0.23694\ N[/tex]

The change in weight is -0.23694 N

The change in your weight if you were to ride an elevator from the street level where you weigh 469N to the top of the building is; -0.237 N

The formula for acceleration here is;

a = GM/r²

Where;

G is gravitational constant = 6.67 × 10⁻¹¹ m³/kg.s²

M is mass of earth = 5.972 × 10²⁴ kg

r is distance from center of earth

Since we are trying to find change in weight, let us first find the change in acceleration with respect to r;

da/dr = -2GM/r³

da =  -(2GM/r³) dr

Thus, change in weight from top to bottom is;

W_top - W_bottom = m(da)

Now, weight at bottom is gotten from the formula;

W_bottom = GmM/r²

Also, W_bottom = m(da) since we are dealing with change in weight.

Thus;

m(da)= -(2GmM/r³) dr

Recall that GmM/r². Thus;

m(da) = -2W_bottom × dr/r

where;

W_bottom = 469 N

r is radius of earth = 6371000 m

dr = 1 mile = 1609.34 m

Thus;

m(da) = -2 × 469 × 1609.34/6371000

m(da) = -0.237 N

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Answer:

a) -6,750 N  b) -1.35*10⁶ N  c) 11.5 times d) 2,295 times

Explanation:

a) In order to answer the question, we can apply Newton's 2nd Law to the person, as follows:

Fnet = m*a

Assuming that the deceleration is uniform, we can use any of the kinematic equations.

In this particular case, examining the givens that we have (the final speed (which is 0), the initial speed (impact speed), and the distance of deceleration (1 m) the most useful equation is the following:

vf²-v₀² = 2*a*d

Replacing by the givens, and solving for a, we get:

a = (15m/s)² / 2* (1 m) = -112.5 m/s²

With this value of a, we can get the net force F:

F = 60 kg* (-112.5 m/s²) = -6,750 N

b) For this part, same reasoning applies, the only difference being the deceleration distance, which for this case is only 5 mm.

We can apply the same kinematic equation:

vf²-v₀² = 2*a*d

Once again, replacing by the givens, and solving for a, we get:

a = (15m/s)² / 2* (0.005 m) = -22,500 m/s²

With this value of a, we can get the net force F:

F = 60 kg* (-22,500 m/s²) = -1.35*10⁶ N

c) If we compare the forces that  we got above with the weight of the person (which is the same to compare the acceleration with g), we have, for the first case (person restrained) a value of  approximately 12 g, but for the unrestrained case, we got a value of 2,295 g!

Answer:

d. 37°C

Explanation:

When Equilibrium is reached,

Heat lost = heat gain.

Heat lost by the hot metal = c₁m₁(T₁-T₂)............. equation 1

Where c₁ = specific heat capacity of the metal = 0.25 cal/g⋅°C, m₁ = mass of the metal = 250 g, T₁ = initial Temperature of the metal = 70°C, T₂ final temperature of the metal .

Substituting this values into equation 1,

Heat lost by the metal = 62.5(70-T₂)

Also,

Heat gain by the water = c₂m₂(T₂-T₁)................. equation 2

c₂ = 1.00 cal/g⋅°C., m₂ = 75 g, T₁ = 20°C

Substituting this values into equation 2,

Heat gain by water = 1× 75 (T₂ - 20)

Heat gained by water = 75(T₂ - 20)

Also,

Heat gained by the calorimeter = c₃m₃(T₂-T₁)............. equation 3

Where c₃ = 0.10 cal/ g⋅°C, m =500 g, T₁ =20°C.

Substituting this values into equation 3

Heat gained by the calorimeter = 0.1 × 500(T₂ - 20)

Heat gained by the calorimeter = 50(T₂ - 20)

Heat lost by the metal = heat gained by water + heat gained by the calorimeter.

62.5(70-T₂) = 50(T₂ - 20) + 75(T₂ - 20)

4375 - 62.5T₂ = 50T₂ - 1000 + 75T₂ - 1500

Collecting like terms,

-62.5T₂ - 50T₂ - 75T₂ =  - 1000 - 1500 - 4375

  -187.5T₂ = -6875

Dividing both side by the coefficient of T₂

-187.5T₂ / -187.5 = -6875 /-187.5

 T₂ = 36.666°C

T₂ ≈ 37°C

The final temperature = 37°C

Answer:

option B

Explanation:

height, h = 145 m

cw = 4186 J/kg °C

g = 9.8 m/s^2

According to the conservation of energy

Potential energy = thermal energy

m x g x h = m x c x ΔT

where, ΔT is the rise in temperature

9.8 x 145 = 4186 x ΔT

ΔT = 0.34°C

Answer:[tex]\Delta T=0.339^{\circ}C[/tex]                    

Explanation:

Given

height from which water is falling [tex]h=145 m[/tex]

heat capacity of water [tex]c_w=4186 J/kg-^{\circ}C[/tex]

here Potential Energy is converted to heat the water

i.e. [tex]P.E.=mc_w\Delta T[/tex]

[tex]mgh=mc_w\Delta T[/tex]

[tex]9.8\times 145=4186\times \Delta T[/tex]

[tex]\Delta T=0.339^{\circ}C[/tex]                      

Answer:

The speed of the particle after travelling for 0.5 m is 100 m/s.                            

Explanation:

It is given that,

Mass of the particle, m = 0.01 kg

Net charge on the particle, q = -0.05 C  

Electric field strength, E = 2000 V/C

Distance travelled by the particle, d = 0.5 m

The work done due to motion of the particle is balanced by the change in kinetic energy as :

[tex]Fd=\dfrac{1}{2}mv^2[/tex]

v is the speed of the particle

F is the electric force

[tex]qEd=\dfrac{1}{2}mv^2[/tex]

[tex]v=\sqrt{\dfrac{2qEd}{m}}[/tex]

[tex]v=\sqrt{\dfrac{2\times 0.05\times 2000\times 0.5}{0.01}}[/tex]

v = 100 m/s

So, the speed of the particle after travelling for 0.5 m is 100 m/s. Hence, this is the required solution.

Answer:

24 hours

[tex]7\times 10^{-5}\ rad/s[/tex]

Explanation:

If a satellite is in sync with Earth then the period of each satellite is 24 hours.

[tex]T=24\times 60\times 60\ s[/tex]

Angular velocity is given by

[tex]\omega=\dfrac{2\pi}{T}\\\Rightarrow \omega=\dfrac{2\pi}{24\times 60\times 60}\\\Rightarrow \omega=7\times 10^{-5}\ rad/s[/tex]

The angular velocity of the satellite is [tex]7\times 10^{-5}\ rad/s[/tex]

A communications satellite in a geostationary orbit must have a period of 24 hours and an angular velocity of approximately 0.262 radians per hour.

To remain above the same point on Earth's surface, a communications satellite must be placed in a geostationary orbit. The period of such a satellite must be 24 hours, which is the same as one Earth day. This means that the satellite completes one orbit around the Earth in 24 hours. The angular velocity of the satellite depends on its position in the orbit and can be calculated using the formula:

Angular velocity = 2π / Period

Since the period is 24 hours, the angular velocity of the satellite is approximately 0.262 radians per hour.

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