Q: A 2 kg rock is dropped from a height of 10 m. Ignoring air resistance, what is the change in gravitational potential energy as it reaches the ground? (Take $$g = 9.8\ \mathrm{m/s^2$})$$

$$\Delta U_{\mathrm{grav}} = m g (y_2$ - y_1$) = 2 \times 9.8 \times (0 - 10) = -196\ \mathrm{J}$$

Q: A spring with spring constant $$k = 200\ \mathrm{N/m} is $$compressed by $$0.1\ \mathrm{m}. $$What is the elastic potential energy stored in the spring?

$$U_{\mathrm{el}} = \frac{1}{2} k x^2$ = \frac{1}{2} \times 200 \times (0.1)^2$ = 1\ \mathrm{J}$$

Q: A 0.5 kg ball is thrown vertically upward with an initial speed of $$10\ \mathrm{m/s}. $$What is its maximum height above the launch point? (Ignore air resistance, $$g = 9.8\ \mathrm{m/s^2$})$$

$$h = \frac{v^2$}{2g} = \frac{10^2$}{2 \times 9.8} \approx 5.1\ \mathrm{m}$$

Question 1

A 2 kg rock is dropped from a height of 10 m. Ignoring air resistance, what is the change in gravitational potential energy as it reaches the ground? (Take $$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$\Delta U_{\mathrm{grav}} = m g (y_2$ - y_1$) = 2 	imes 9.8 	imes (0 - 10) = -196\ \mathrm{J}$$

Question 2

A spring with spring constant $$k = 200\ \mathrm{N/m} is $$compressed by $$0.1\ \mathrm{m}. $$What is the elastic potential energy stored in the spring?

Accepted Answer

$$U_{\mathrm{el}} = \frac{1}{2} k x^2$ = \frac{1}{2} 	imes 200 	imes (0.1)^2$ = 1\ \mathrm{J}$$

Question 3

Which of the following statements about conservative forces is correct?

Accepted Answer

The work done by a conservative force is independent of the path taken between two points.

Question 4

A 0.5 kg ball is thrown vertically upward with an initial speed of $$10\ \mathrm{m/s}. $$What is its maximum height above the launch point? (Ignore air resistance, $$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$h = \frac{v^2$}{2g} = \frac{10^2$}{2 	imes 9.8} \approx 5.1\ \mathrm{m}$$

Question 5

A block slides down a frictionless incline and compresses a spring at the bottom. If the block starts from rest at height $$h$$ and the spring constant is $$k$$, what is the maximum compression $$x of $$the spring?

Accepted Answer

$$x = \sqrt{\frac{2 m g h}{k}}$$

Question 6

Which of the following is NOT a property of a conservative force?

Accepted Answer

The work done depends on the path taken.

Question 7

A car tire heats up after driving due to internal friction. What type of force is responsible for this energy transformation?

Accepted Answer

Nonconservative force

Question 8

Given the potential energy function $$U(x) = 4x^2$, what is the corresponding force F_x(x$$)$$?

Accepted Answer

$$F_x(x) = -\frac{dU}{dx} = -8x$$

Question 9

A hiker climbs from the base to the top of a mountain. Which statement is true about the work done by gravity?

Accepted Answer

The work done by gravity depends only on the vertical displacement, not the path taken.

Question 10

A glider attached to a spring oscillates on a frictionless air track. At the turning points of its motion, which of the following is true?

Accepted Answer

The kinetic energy is zero and the potential energy is maximum.

Question 11

Which equation correctly expresses the law of conservation of mechanical energy for a system with only gravity acting?

Accepted Answer

$$K_1 + U$$_{\mathrm{grav},1} = $$K_2 + U$$_{\mathrm{grav},2}$$

Question 12

A mass is attached to a spring and oscillates horizontally. If the spring is compressed by $$x$$ from equilibrium, what is the direction of the force exerted by the spring?

Accepted Answer

Toward the equilibrium position

Question 13

Which of the following best describes stable equilibrium in terms of potential energy?

Accepted Answer

A point where the potential energy is at a minimum

Question 14

A parachutist descends at constant speed. Which statement is true about the forces and energy involved?

Accepted Answer

Air resistance does negative work, and the mechanical energy decreases.

Question 15

If $$U(x) is $$the potential energy function, what is the general formula for the force in one dimension?

Accepted Answer

$$F_x(x) = -\frac{dU}{dx}$$

Question 16

A gymnast balances on a unicycle. What type of equilibrium is she demonstrating?

Accepted Answer

Unstable equilibrium

Question 17

Which mathematical operation gives the components of a conservative force in three dimensions from a potential energy function $$U(x, y, z)$$?

Accepted Answer

Taking the negative partial derivatives: $$F_x = -\frac{\partial U}{\partial x}$$, $$F_y = -\frac{\partial U}{\partial y}$$, $$F_z = -\frac{\partial U}{\partial z}$$

Ch7: Potential Energy and Energy Conservation

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