Q: A 2 kg ball is dropped from a height of 100 m. What is its total mechanical energy at the top of the building? (Assume $$g = 9.8 \ \mathrm{m/s^2$}$$ and neglect air resistance.)

$$E = mgh = 2 \times 9.8 \times 100 = 1960 \ \mathrm{J}$$

Q: A 4 kg object is launched vertically upward from the ground with an initial speed of 40 m/s. What is the maximum height it reaches? (Ignore air resistance, $$g = 9.8 \ \mathrm{m/s^2$})$$

$$h = \frac{v^2$}{2g} = \frac{40^2$}{2 \times 9.8} = 81.6 \ \mathrm{m}$$

Q: A 6 kg object is thrown downward from a height of 20 m and reaches the ground with a speed of 30 m/s. What was its initial speed? (Use $$g = 9.8 \ \mathrm{m/s^2$})$$

$$v_0$$ = \sqrt{$$v^2 - 2g$$\Delta y} = \sqrt{$$30^2 - 2$$ \times 9.8 \times 20} = 10 \ \mathrm{m/s}$$

Q: A block of mass $$m$$ slides along a flat surface with an initial speed of 30 m/s and stops due to friction ($$\mu_k = 0.6). $$What distance does it travel before stopping? (Use $$g = 9.8 \ \mathrm{m/s^2$})$$

$$d = \frac{v^2$}{2 \mu_k g} = \frac{30^2$}{2 \times 0.6 \times 9.8} = 76.5 \ \mathrm{m}$$

Q: A 4 kg block moving at 20 m/s strikes a massless spring ($$k = 600 \ \mathrm{N/m}) on a $$frictionless surface. What is the maximum compression of the spring?

$$\Delta x = \sqrt{\frac{mv^2$}{k}} = \sqrt{\frac{4 \times 400}{600}} = 1.63 \ \mathrm{m}$$

Question 1

Which of the following best describes the difference between conservative and non-conservative forces?

Accepted Answer

Conservative forces do not change the total mechanical energy of a system, while non-conservative forces cause a change in the total mechanical energy.

Question 2

A 2 kg ball is dropped from a height of 100 m. What is its total mechanical energy at the top of the building? (Assume $$g = 9.8 \ \mathrm{m/s^2$}$$ and neglect air resistance.)

Accepted Answer

$$E = mgh = 2 	imes 9.8 	imes 100 = 1960 \ \mathrm{J}$$

Question 3

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

Accepted Answer

$$K_i + U_i = K_f + U_f$$

Question 4

A 4 kg object is launched vertically upward from the ground with an initial speed of 40 m/s. What is the maximum height it reaches? (Ignore air resistance, $$g = 9.8 \ \mathrm{m/s^2$})$$

Accepted Answer

$$h = \frac{v^2$}{2g} = \frac{40^2$}{2 	imes 9.8} = 81.6 \ \mathrm{m}$$

Question 5

A 6 kg object is thrown downward from a height of 20 m and reaches the ground with a speed of 30 m/s. What was its initial speed? (Use $$g = 9.8 \ \mathrm{m/s^2$})$$

Accepted Answer

$$v_0$$ = \sqrt{$$v^2 - 2g$$\Delta y} = \sqrt{$$30^2 - 2$$ 	imes 9.8 	imes 20} = 10 \ \mathrm{m/s}$$

Question 6

Which of the following is NOT a form of mechanical energy?

Accepted Answer

Thermal energy

Question 7

A 0.5 kg hockey puck is moving at 4 m/s. You push it with a constant 200 N force over 0.3 m. What is its final speed? (Neglect friction)

Accepted Answer

$$v_f = \sqrt{v_i^2$ + \frac{2F d}{m}} = \sqrt{4^2$ + \frac{2 	imes 200 	imes 0.3}{0.5}} = 16 \ \mathrm{m/s}$$

Question 8

A block of mass $$m$$ slides along a flat surface with an initial speed of 30 m/s and stops due to friction ($$\mu_k = 0.6). $$What distance does it travel before stopping? (Use $$g = 9.8 \ \mathrm{m/s^2$})$$

Accepted Answer

$$d = \frac{v^2$}{2 \mu_k g} = \frac{30^2$}{2 	imes 0.6 	imes 9.8} = 76.5 \ \mathrm{m}$$

Question 9

A 2 kg object is dropped from 80 m and reaches the ground at 30 m/s. What is the work done by air resistance?

Accepted Answer

$$W_{air} = \Delta K + \Delta U = \frac{1}{2} m v^2$ - mgh = 0.5 	imes 2 	imes 30^2$ - 2 	imes 9.8 	imes 80 = 900 - 1568 = -668 \ \mathrm{J}$$

Question 10

A 4 kg block moving at 20 m/s strikes a massless spring ($$k = 600 \ \mathrm{N/m}) on a $$frictionless surface. What is the maximum compression of the spring?

Accepted Answer

$$\Delta x = \sqrt{\frac{mv^2$}{k}} = \sqrt{\frac{4 	imes 400}{600}} = 1.63 \ \mathrm{m}$$

Question 11

Which of the following statements about potential energy graphs is correct?

Accepted Answer

The kinetic energy at any point is the difference between the total mechanical energy and the potential energy at that point.

Question 12

On a $$U(x)$$ graph, at which point is the force on the object zero?

Accepted Answer

Where the slope of $$U(x) is $$zero (horizontal tangent)

Question 13

A 3 kg block hangs 2 m above the floor, connected to a 4 kg block on a rough table ($$\mu_k = 0.5). If $$released from rest, what is the speed of the 3 kg block just before hitting the floor? (Neglect pulley/string mass, $$g = 9.8 \ \mathrm{m/s^2$})$$

Accepted Answer

$$v = \sqrt{\frac{2(m_1$ g h - \mu_k m_2$ g h)}{m_1$ + m_2$}} = \sqrt{\frac{2(3 	imes 9.8 	imes 2 - 0.5 	imes 4 	imes 9.8 	imes 2)}{3+4}} = 3.13 \ \mathrm{m/s}$$

Question 14

A 30 kg block is connected to a 20 kg block on a smooth 53° incline. If released from rest, what is the speed of the system after the 30 kg block has descended 1 m? (Use $$g = 9.8 \ \mathrm{m/s^2$})$$

Accepted Answer

$$v = \sqrt{\frac{2(30 	imes 9.8 	imes 1 - 20 	imes 9.8 	imes 1 	imes \sin 53^\circ)}{30+20}} = 1.98 \ \mathrm{m/s}$$

Question 15

Which of the following is true about stable equilibrium points on a potential energy graph?

Accepted Answer

They occur at local minima of $$U(x)$$, and if displaced slightly, the object returns to equilibrium.

Question 16

A 4 kg block moving at 20 m/s strikes a spring ($$k = 600 \ \mathrm{N/m}) on a $$surface with friction ($$\mu_k = 0.5). $$What is the maximum compression of the spring?

Accepted Answer

$$\Delta x = \sqrt{\frac{mv^2$}{k + 2\mu_k m g}} = \sqrt{\frac{4 	imes 400}{600 + 2 	imes 0.5 	imes 4 	imes 9.8}} = 1.38 \ \mathrm{m}$$

Question 17

A particle of mass 0.1 kg moves according to a potential energy graph. What minimum speed does it need at point A (where $$U = 4 \ \mathrm{J}) to $$reach point B (where $$U = 2 \ \mathrm{J}$$), if the total mechanical energy must be at least $$2 \ \mathrm{J} at B$$?

Accepted Answer

$$v = \sqrt{\frac{2(E - U_A)}{m}} = \sqrt{\frac{2(2 - 4)}{0.1}} = 	ext{Not possible, needs at least 4 J}$$

Potential Energy and Conservation: Study Notes

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