Q: A roller coaster of mass 100 kg starts from rest at a height of 40 m. If there is no friction, what is its speed at a point where its height is 30 m? (Use $$g = 9.8 \mathrm{\ m/s^2$})$$

$$v = \sqrt{2g(h_A - h_B)} = \sqrt{2 \times 9.8 \times (40 - 30)} = 14 \mathrm{\ m/s}$$

Q: A 0.100 kg dart is pressed against the spring of a toy dart gun (spring constant $$k = 250 \mathrm{\ N/m}) $$and compressed $$6.0 \mathrm{\ cm}. $$What speed does the dart acquire when released? (Assume no friction.)

$$v = \sqrt{\frac{kx^2$}{m}} = \sqrt{\frac{250 \times (0.06)^2$}{0.1}} = 3.0 \mathrm{\ m/s}$$

Q: You do $$174 \mathrm{\ J} of $$work pulling your sister back on a swing (chain length $$5.10 \mathrm{\ m}) $$until the chain makes an angle of $$32.0^\circ$$ with the vertical. What is your sister's mass? (Assume $$g = 9.8 \mathrm{\ m/s^2$}$$ and negligible friction.)

$$m = \frac{174}{9.8 \times (5.10 - 5.10 \cos 32^\circ)} \approx 23.9 \mathrm{\ kg}$$

Q: A child on a swing has a speed of $$7.7 \mathrm{\ m/s} at $$the lowest point. How high will the swing be at the high point?

$$h = \frac{v^2$}{2g} = \frac{(7.7)^2$}{2 \times 9.8} \approx 3.0 \mathrm{\ m}$$

Q: An object of mass $$10.0 \mathrm{\ kg} is $$released from rest at the top of a frictionless inclined plane of height $$8.00 \mathrm{\ m}. $$What is its kinetic energy at the bottom?

$$K = mgh = 10.0 \times 9.8 \times 8.00 = 784 \mathrm{\ J}$$

Q: A slingshot fires a pebble from a building that is $$31.0 \mathrm{\ m}$$ tall with a speed of $$14.0 \mathrm{\ m/s}. $$What is the speed of the pebble just before it hits the ground?

$$v = \sqrt{v_0$^2 + 2gh} = \sqrt{(14.0)^2$ + 2 \times 9.8 \times 31.0} \approx 28.0 \mathrm{\ m/s}$$

Question 1

A roller coaster of mass 100 kg starts from rest at a height of 40 m. If there is no friction, what is its speed at a point where its height is 30 m? (Use $$g = 9.8 \mathrm{\ m/s^2$})$$

Accepted Answer

$$v = \sqrt{2g(h_A - h_B)} = \sqrt{2 	imes 9.8 	imes (40 - 30)} = 14 \mathrm{\ m/s}$$

Question 2

Which of the following statements about conservative and nonconservative forces is TRUE?

Accepted Answer

Work done by a conservative force on any closed path is zero.

Question 3

A 0.100 kg dart is pressed against the spring of a toy dart gun (spring constant $$k = 250 \mathrm{\ N/m}) $$and compressed $$6.0 \mathrm{\ cm}. $$What speed does the dart acquire when released? (Assume no friction.)

Accepted Answer

$$v = \sqrt{\frac{kx^2$}{m}} = \sqrt{\frac{250 	imes (0.06)^2$}{0.1}} = 3.0 \mathrm{\ m/s}$$

Question 4

You do $$174 \mathrm{\ J} of $$work pulling your sister back on a swing (chain length $$5.10 \mathrm{\ m}) $$until the chain makes an angle of $$32.0^\circ$$ with the vertical. What is your sister's mass? (Assume $$g = 9.8 \mathrm{\ m/s^2$}$$ and negligible friction.)

Accepted Answer

$$m = \frac{174}{9.8 	imes (5.10 - 5.10 \cos 32^\circ)} \approx 23.9 \mathrm{\ kg}$$

Question 5

A hockey puck slides on smooth ice at $$4 \mathrm{\ m/s}$$ and comes to a $$1 \mathrm{\ m}$$ high hill. Will it make it to the top?

Accepted Answer

No, because $$\frac{1}{2}mv^2$ = mgh$$ requires $$v^2 = 2gh = 19.6$, but v^2$ = 16.$$

Question 6

Which equation correctly expresses the conservation of mechanical energy for a system with only conservative forces?

Accepted Answer

$$\Delta E_{\mathrm{mech}} = \Delta K + \Delta U = 0$$

Question 7

A skier drops a $$0.312 \mathrm{\ kg}$$ glove from rest into fresh snow. The glove falls to a depth $$d = 0.147 \mathrm{\ m}$$ below the surface before stopping. If the nonconservative work done is $$W_{\mathrm{nc}} = -1.44 \mathrm{\ J}$$, what was the drop height?

Accepted Answer

$$h = 0.32 \mathrm{\ m}$$

Question 8

Which of the following is NOT a conservative force?

Accepted Answer

Friction

Question 9

A $$2.0 \mathrm{\ kg}$$ package slides along a floor at $$v_1 = 4.0$$ \mathrm{\ m/s}$$ and compresses a spring (k = 10000$$ \mathrm{\ N/m}$$) until it stops. If a 15$$ \mathrm{\ N}$$ friction force acts, what is the spring's compression d$ when the package stops?

Accepted Answer

$$d = 0.08 \mathrm{\ m}$$

Question 10

Which statement best describes the effect of nonconservative forces on a system's mechanical energy?

Accepted Answer

They convert mechanical energy into other forms, such as heat.

Question 11

A child on a swing has a speed of $$7.7 \mathrm{\ m/s} at $$the lowest point. How high will the swing be at the high point?

Accepted Answer

$$h = \frac{v^2$}{2g} = \frac{(7.7)^2$}{2 	imes 9.8} \approx 3.0 \mathrm{\ m}$$

Question 12

Which formula gives the potential energy stored in a spring?

Accepted Answer

$$U = \frac{1}{2}kx^2$

Question 13

An object of mass $$10.0 \mathrm{\ kg} is $$released from rest at the top of a frictionless inclined plane of height $$8.00 \mathrm{\ m}. $$What is its kinetic energy at the bottom?

Accepted Answer

$$K = mgh = 10.0 	imes 9.8 	imes 8.00 = 784 \mathrm{\ J}$$

Question 14

A slingshot fires a pebble from a building that is $$31.0 \mathrm{\ m}$$ tall with a speed of $$14.0 \mathrm{\ m/s}. $$What is the speed of the pebble just before it hits the ground?

Accepted Answer

$$v = \sqrt{v_0$^2 + 2gh} = \sqrt{(14.0)^2$ + 2 	imes 9.8 	imes 31.0} \approx 28.0 \mathrm{\ m/s}$$

Question 15

Which of the following is a correct statement of the law of conservation of energy?

Accepted Answer

Energy is never created or destroyed; it only changes form.

Question 16

A $$20 \mathrm{\ kg}$$ object has a velocity of $$2 \mathrm{\ m/s} at $$the top of a $$200 \mathrm{\ m}$$ hill. At the bottom, its velocity is $$10 \mathrm{\ m/s}. $$How much work was done by friction?

Accepted Answer

$$W_{\mathrm{friction}} = E_f - E_i = [\frac{1}{2} 	imes 20 	imes 10^2$ + 0] - [\frac{1}{2} 	imes 20 	imes 2^2$ + 20 	imes 9.8 	imes 200] = 1000 - 39240 = -38240 \mathrm{\ J}$$

Question 17

Which of the following is TRUE about the work done by a conservative force?

Accepted Answer

It is the negative of the change in potential energy.

Conservation of Mechanical Energy and Energy Principles

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