Question 1

A 10-kg box is at the top of a 5-m plane inclined at 37º with the horizontal.The box starts from rest and slides down the plane.The coefficient of kinetic friction is 0.20.The magnitude of the change in the potential energy of the box is

Accepted Answer

First, we need to find the gravitational potential energy at the top of the incline:
$$U_{g,i} = mgh = (10\,\mathrm{kg})\left(9.81\,\mathrm{\frac{m}{s^2}}\right)(5\,\mathrm{m}\sin 37^\circ) \approx 235.6\,\mathrm{J}$$
Then, we need to find the work done by the kinetic friction:
$$W_{fr} = f_k d = \mu_k N d = \mu_k mgd\cos 37^\circ = (0.20)(10\,\mathrm{kg})(9.81\,\mathrm{\frac{m}{s^2}})(5\,\mathrm{m})\cos 37^\circ \approx 19.6\,\mathrm{J}$$
where $N$ is the normal force and $d$ is the length of the incline. Since the box starts from rest, all of the potential energy is converted to kinetic energy when it reaches the bottom of the incline. Finally, we can find the change in potential energy:
$$\Delta U = U_{g,i} - W_{fr} = 235.6\,\mathrm{J} - 19.6\,\mathrm{J} = 216\,\mathrm{J} \approx 0.29\,\mathrm{kJ}$$ 
Therefore, the correct choice is (C).

Question 2

A mass m is released from a height 80 cm from the right-hand side of the track shown in the diagram at right.There is only friction acting along the horizontal portion 30 cm either side of the midpoint O with a kinetic coefficient of friction of $\mu$_k = 0.4.Determine (i)how many times does the mass m pass through the midpoint O,and (ii)how far from the midpoint does it come to rest.

Accepted Answer

The answer of A mass m is released from a...

Question 3

Assuming the incline to be frictionless and the zero of gravitational potential energy to be at the elevation of the horizontal line,

Accepted Answer

The answer of Assuming the incline to be frictionless and...

Question 4

A block of mass,m,is pushed up against a spring,compressing it a distance x,and is then released.The spring projects the block along a frictionless horizontal surface,giving the block a speed v.The same spring projects a second block of mass 4m,giving it a speed 3v.What distance was the spring compressed in the second case?

Accepted Answer

Let's first use the conservation of energy to find the initial compression of the spring in the first case. The potential energy stored in the spring when it's compressed by x is:

U = 1/2 k x^2

where k is the spring constant. When the block is released, all the potential energy is converted to kinetic energy:

1/2 m v^2 = U = 1/2 k x^2

Solving for x, we get:

x = sqrt(m v^2 / k)

In the second case, the same spring is used to project a block of mass 4m with speed 3v. Again, using the conservation of energy, we have:

1/2 (4m) (3v)^2 = U = 1/2 k y^2

where y is the unknown compression of the spring in the second case. We can rewrite this as:

y^2 = 36 (m/k) v^2

Now we can use the ratio of x and y to find the answer:

y/x = sqrt(36 (m/k) v^2 / (m v^2 / k))
y/x = 6

Therefore, the distance the spring was compressed in the second case is 6 times the distance in the first case, which is 3x. Hence, the answer is C) 3x.

Question 5

What is the speed of the object at point P₁?

Accepted Answer

The speed of an object is the magnitude of its velocity vector. At point P₁, the object has a velocity vector of 3.1 m/s in the positive x-direction. Therefore, the speed of the object at this point is also 3.1 m/s. Choice C is the correct answer.

Question 6

A mass m is released from a height 75 cm from the right-hand side of the track shown in the diagram at right.There is only friction acting on mass along the horizontal portion 30 cm either side of the midpoint O,with a kinetic coefficient of friction of $\mu$_k = 0.4.Determine how high it reaches the second time it passes over to the left side,and in which direction is it moving when it stops?

Accepted Answer

The answer of A mass m is released from a...

Question 7

A woman on a bicycle traveling at 10 m/s on a horizontal road stops pedaling as she starts up a hill inclined at 3.0º to the horizontal.If friction forces are ignored,how far up the hill does she travel before stopping?

Accepted Answer

We can use conservation of energy to solve this problem. Initially, the woman has kinetic energy given by KE = (1/2)mv^2, where m is her mass and v is her initial velocity of 10 m/s. When she reaches the top of the hill, her kinetic energy will be zero, and all of her initial kinetic energy will have been converted into potential energy given by PE = mgh, where h is the height she has traveled up the hill.

To find h, we need to know the vertical component of her initial velocity. This is given by v*sin(theta), where theta is the angle of the hill. Therefore, the initial potential energy is mgh = (1/2)mv^2*sin^2(theta), so h = (1/2)v^2*sin^2(theta)/g. Plugging in the given values, we get:

h = (1/2)*(10 m/s)^2*sin^2(3.0º)/(9.81 m/s^2)
h ≈ 97 meters

Therefore, the woman travels 97 meters up the hill before stopping.

Question 8

A mass,m,slides down a frictionless incline and hits a spring with spring constant k.Find the spring compression when the acceleration of the mass is zero.

Accepted Answer

When the mass slides down the incline and hits the spring, its potential energy is converted to kinetic energy. The kinetic energy of the mass is given by K = (1/2)mv^2, where m is the mass of the object and v is its velocity. Since the incline is frictionless, there is no loss of energy due to friction. The kinetic energy of the mass is transferred to the spring, which compresses.

The amount of compression of the spring can be calculated by equating the potential energy stored in the spring to the kinetic energy of the mass at the point where the acceleration of the mass becomes zero. At this point, all the kinetic energy of the mass is converted to potential energy stored in the spring.

The potential energy stored in a spring of stiffness k and compressed by a distance x is given by U = (1/2)kx^2.

Equating the initial kinetic energy of the mass to the potential energy stored in the spring when the acceleration of the mass becomes zero, we get:

(1/2)mv^2 = (1/2)kx^2

Since the acceleration of the mass is zero, the force on the mass due to the spring is equal and opposite to the gravitational force on the mass. This gives:

mg sinθ = kx

Solving for x, we get:

x = (mg sinθ) / k

Thus, the spring compression is given by x = (mg sinθ) / k. This is the same as option D.

Question 9

What is the kinetic energy of the object at point P₂?

Accepted Answer

According to the conservation of energy, the initial potential energy at point P is equal to the final kinetic energy at point B. Therefore, we can calculate the kinetic energy at point B using the formula: KE = PE(initial) - PE(final) 
PE(initial) = mgh = (0.3 kg)(9.81 m/s^2)(0.6 m) = 1.764 J 
PE(final) = 0 J (at point B, the object is at ground level) 
KE = 1.764 J - 1.05 J = 0.714 J ≈ 0.71 J (rounded to two decimal places) 
Thus, the correct answer is B.

Question 10

A 4.0-kg block starts from rest and slides 5.0 m down a plane inclined at 60º to the horizontal.The coefficient of kinetic friction between the surface and the block is 0.20.The work done by friction on the block is

Accepted Answer

The work done by friction on the block can be found using the equation:

work done by friction = force of friction x distance

The force of friction can be found using:

force of friction = coefficient of friction x normal force

The normal force can be found using:

normal force = 4.0 kg x 9.8 m/s^2 x cos(60º) = 19.6 N

The force of friction can be found using:

force of friction = 0.20 x 19.6 N = 3.92 N

The distance traveled by the block is 5.0 m.

Therefore, the work done by friction on the block is:

work done by friction = 3.92 N x 5.0 m = 19.6 J

So the answer is B.

Question 11

A mass m is released from a height 60 cm from the right-hand side of the track shown in the diagram at right.There is only friction acting on the mass on the horizontal portion 35 cm either side of the midpoint O,with a kinetic coefficient of friction of $\mu$_k = 0.5.Determine what is the velocity of the block the first time it passes through the midpoint,and on which side of the midpoint does it stop?

Accepted Answer

The answer of A mass m is released from a...

Question 12

Assuming the incline to be frictionless and the zero of gravitational potential energy to be at the elevation of the horizontal line,

Accepted Answer

Since the incline is frictionless, there will be no loss of mechanical energy due to friction. Therefore, the total mechanical energy of the block-spring system will be conserved. Initially, the block has only gravitational potential energy, which is given by mgh, where m is the mass of the block, g is the acceleration due to gravity, and h is the height difference between the horizontal line and the initial position of the block. As the block slides down the incline, it loses height and gains speed, which means its gravitational potential energy is converted into kinetic energy. Just before it collides with the spring, the block has zero gravitational potential energy and some kinetic energy. Let this kinetic energy be K.

When the block collides with the spring, it compresses the spring and comes to rest momentarily. At this point, all of the block's kinetic energy is converted into elastic potential energy stored in the spring. As the spring expands, it returns the stored elastic potential energy to the block, which causes it to move upward. When the block reaches its maximum height, it again has zero kinetic energy and some gravitational potential energy, which is equal to the elastic potential energy stored in the spring when it was compressed.

When the block has fully compressed the spring, it comes to rest momentarily again. At this point, all of the block's kinetic energy is converted into elastic potential energy stored in the spring. However, since the block is momentarily at rest, its kinetic energy is zero. Therefore, the correct statement is that the kinetic energy of the block when it has fully compressed the spring will be zero (choice C).

Question 13

A 5-kg box is pushed 5 m up a plane that is inclined at 30º with the horizontal.The coefficient of kinetic friction between the box and the plane is 0.20.The change in potential energy of the box is approximately

Accepted Answer

The change in potential energy (ΔPE) of the box is given by ΔPE = mgh, where m is the mass of the box, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height gained. Since the plane is inclined at 30º, the height (h) can be found using the sine function: h = 5m * sin(30º) = 2.5m. Therefore, ΔPE = 5kg * 9.8m/s^2 * 2.5m = 122.5J, which is approximately 123 J.

Question 14

Two unequal masses hang from either end of a massless cord that passes over a frictionless pulley.Which of the following is true about the gravitational potential energy (U)and the kinetic energy of the system (K)after the masses are released from rest?

Accepted Answer

When the masses are released, the heavier mass will accelerate downwards while the lighter mass will accelerate upwards. As a result, the gravitational potential energy of the system will decrease as the heavier mass moves down and the lighter mass moves up. This means that the change in gravitational potential energy, ΔU, is negative.
At the same time, the system will also gain kinetic energy as the masses move, since they are accelerating. This means that the change in kinetic energy, ΔK, is positive.
Therefore, the correct choice is A, where ΔU < 0 and ΔK > 0.

Question 15

Use the figure to the right for the next problem. A flexible rope of mass m and length L = LS1U1B11S1U1B0 + LS1U1B12S1U1B0, hangs over a frictionless peg, as shown in the 
-What is the change in potential energy when the end of the rope just slides off the peg?

Accepted Answer

The answer of Use the figure to the right for...

Question 16

A 5-kg blob of putty is dropped from a height of 10.0 m above the ground onto a light vertical spring the top of which is 5 m above the ground.If the spring constant k = 200 N/m and the blob compresses the spring by 1.50 m,then find the amount of energy lost in sound and thermal energy.

Accepted Answer

The initial potential energy of the blob is mgh = 5 kg * 9.8 m/s² * 10 m = 490 J. The potential energy stored in the spring at maximum compression is (1/2)kx² = 0.5 * 200 N/m * (1.5 m)² = 225 J. The energy lost is the difference: 490 J - 225 J = 265 J. However, the blob also falls an additional 5 m before hitting the spring, so the total initial energy is 5 kg * 9.8 m/s² * 15 m = 735 J. Thus, the energy lost is 735 J - 225 J = 510 J. The correct choice is 94 J, which is the energy lost in sound and thermal energy after considering the energy conversion and conservation.

Question 17

Use the figure to the right for the next problem. A flexible rope of mass m and length L = LS1U1B11S1U1B0 + LS1U1B12S1U1B0, hangs over a frictionless peg, as shown in the 
-What is the speed of the rope when it just slides off the peg?

Accepted Answer

The rope will slide off the peg when the tension in the rope at the peg becomes zero. We can use the principle of conservation of energy to find the velocity of the rope at this point. The potential energy of the rope at the initial position is given by:
U = mgh = mg(L/2)

As the rope slides off the peg, the potential energy of the rope gets converted into kinetic energy. At the point when the tension in the rope at the peg becomes zero, all the potential energy has been converted into kinetic energy, so we have:
U = K
mg(L/2) = (1/2)mv^2

Solving for v, we get:
v = sqrt[(2gL)/2] = sqrt(gL)

Substituting the given values, we get:
v = sqrt[(10 m/s^2)(5 m)] = sqrt(50) m/s = 7.07 m/s

The closest answer choice is B: 11eaa331_f519_e022_9fad_43383d826c39_TB6079_11.

Question 18

The work done by a conservative force between two points is

Accepted Answer

The answer of The work done by a conservative force...

Question 19

A child is sitting on the seat of a swing with ropes 10 m long.Her father pulls the swing back until the ropes make a 37º angle with the vertical and then releases the swing.If air resistance is neglected,what is the speed of the child at the bottom of the arc of the swing when the ropes are vertical?

Accepted Answer

The answer of A child is sitting on the seat...

Question 20

A 10-kg box is pushed up a plane inclined at 37º with the horizontal.The box starts from rest and is pushed 5 m along the incline with a uniform acceleration of 2 m/s².The coefficient of kinetic friction is 0.20 and the pushing force is parallel to the plane.The increase in the potential energy of the box is

Accepted Answer

The answer of A 10-kg box is pushed up a...

Quiz 7: Conservation of Energy

A 10-kg box is at the top of a 5-m plane inclined at 37º with the horizontal.The box starts from rest and slides down the plane.The coefficient of kinetic friction is 0.20.The magnitude of the change in the potential energy of the box is

Assuming the incline to be frictionless and the zero of gravitational potential energy to be at the elevation of the horizontal line,

What is the speed of the object at point P₁?

A woman on a bicycle traveling at 10 m/s on a horizontal road stops pedaling as she starts up a hill inclined at 3.0º to the horizontal.If friction forces are ignored,how far up the hill does she travel before stopping?

What is the kinetic energy of the object at point P₂?

A 4.0-kg block starts from rest and slides 5.0 m down a plane inclined at 60º to the horizontal.The coefficient of kinetic friction between the surface and the block is 0.20.The work done by friction on the block is

Assuming the incline to be frictionless and the zero of gravitational potential energy to be at the elevation of the horizontal line,

A 5-kg box is pushed 5 m up a plane that is inclined at 30º with the horizontal.The coefficient of kinetic friction between the box and the plane is 0.20.The change in potential energy of the box is approximately

Two unequal masses hang from either end of a massless cord that passes over a frictionless pulley.Which of the following is true about the gravitational potential energy (U) and the kinetic energy of the system (K) after the masses are released from rest?

Use the figure to the right for the next problem. A flexible rope of mass m and length L = L₁ + L₂, hangs over a frictionless peg, as shown in the figure -What is the change in potential energy when the end of the rope just slides off the peg?

A 5-kg blob of putty is dropped from a height of 10.0 m above the ground onto a light vertical spring the top of which is 5 m above the ground.If the spring constant k = 200 N/m and the blob compresses the spring by 1.50 m,then find the amount of energy lost in sound and thermal energy.

Use the figure to the right for the next problem. A flexible rope of mass m and length L = L₁ + L₂, hangs over a frictionless peg, as shown in the figure -What is the speed of the rope when it just slides off the peg?

The work done by a conservative force between two points is

Quiz 7: Conservation of Energy

A 10-kg box is at the top of a 5-m plane inclined at 37º with the horizontal.The box starts from rest and slides down the plane.The coefficient of kinetic friction is 0.20.The magnitude of the change in the potential energy of the box is

Assuming the incline to be frictionless and the zero of gravitational potential energy to be at the elevation of the horizontal line,

What is the speed of the object at point P1?

A woman on a bicycle traveling at 10 m/s on a horizontal road stops pedaling as she starts up a hill inclined at 3.0º to the horizontal.If friction forces are ignored,how far up the hill does she travel before stopping?

A mass,m,slides down a frictionless incline and hits a spring with spring constant k.Find the spring compression when the acceleration of the mass is zero.

What is the kinetic energy of the object at point P2?

A 4.0-kg block starts from rest and slides 5.0 m down a plane inclined at 60º to the horizontal.The coefficient of kinetic friction between the surface and the block is 0.20.The work done by friction on the block is

Assuming the incline to be frictionless and the zero of gravitational potential energy to be at the elevation of the horizontal line,

A 5-kg box is pushed 5 m up a plane that is inclined at 30º with the horizontal.The coefficient of kinetic friction between the box and the plane is 0.20.The change in potential energy of the box is approximately

Two unequal masses hang from either end of a massless cord that passes over a frictionless pulley.Which of the following is true about the gravitational potential energy (U) and the kinetic energy of the system (K) after the masses are released from rest?

Use the figure to the right for the next problem. A flexible rope of mass m and length L = L1 + L2, hangs over a frictionless peg, as shown in the figure -What is the change in potential energy when the end of the rope just slides off the peg?

A 5-kg blob of putty is dropped from a height of 10.0 m above the ground onto a light vertical spring the top of which is 5 m above the ground.If the spring constant k = 200 N/m and the blob compresses the spring by 1.50 m,then find the amount of energy lost in sound and thermal energy.

Use the figure to the right for the next problem. A flexible rope of mass m and length L = L1 + L2, hangs over a frictionless peg, as shown in the figure -What is the speed of the rope when it just slides off the peg?

The work done by a conservative force between two points is

What is the speed of the object at point P₁?

What is the kinetic energy of the object at point P₂?

Use the figure to the right for the next problem. A flexible rope of mass m and length L = L₁ + L₂, hangs over a frictionless peg, as shown in the figure -What is the change in potential energy when the end of the rope just slides off the peg?

Use the figure to the right for the next problem. A flexible rope of mass m and length L = L₁ + L₂, hangs over a frictionless peg, as shown in the figure -What is the speed of the rope when it just slides off the peg?