Q: A ball of mass $$m is $$dropped from a height $$y_0$$ and falls freely under gravity. Which of the following correctly expresses the conservation of mechanical energy for the ball during its fall?

$$E_{mech} = U_g + K = mgy + \frac{1}{2}mv^2$ = \text{constant}$$

Question 1

A ball of mass $$m is $$dropped from a height $$y_0$$ and falls freely under gravity. Which of the following correctly expresses the conservation of mechanical energy for the ball during its fall?

Accepted Answer

$$E_{mech} = U_g + K = mgy + \frac{1}{2}mv^2$ = 	ext{constant}$$

Question 2

For a mass-spring system with spring constant $$k$$ and equilibrium position $$x_0$$, what is the expression for the potential energy stored in the spring when the mass is at position $$x$$?

Accepted Answer

$$U_{sp} = \frac{1}{2}k(x - x_0$)^2$

Question 3

A particle is placed at a position corresponding to a local minimum of the potential energy curve $$U(x). $$What type of equilibrium does this represent, and what happens if the particle is given a small 'kick'?

Accepted Answer

Stable equilibrium; the particle oscillates about the minimum.

Question 4

Which mathematical condition must be satisfied at an equilibrium position $$x_{eq} of a $$potential energy function $$U(x)$$?

Accepted Answer

$$\left. \frac{dU}{dx} \right|_{x_{eq}} = 0$$

Question 5

A force $$F_s is $$related to the potential energy $$U by $$which of the following expressions?

Accepted Answer

$$F_s = -\frac{dU}{ds}$$

Question 6

For a spring obeying Hooke's law, what is the force exerted by the spring when it is stretched by a displacement $$\Delta x$$ from equilibrium?

Accepted Answer

$$F_{sp} = -k \Delta x$$

Question 7

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 8

A particle moves from point A to point B under the influence of a force $$\vec{F}. If $$the work done by $$\vec{F}$$ depends on the path taken, what can be concluded about $$\vec{F}$$?

Accepted Answer

$$\vec{F} is a $$nonconservative force.

Question 9

Consider a potential energy curve $$U(x)$$ with a local maximum at $$x_3. If a $$particle is placed at $$x_3$$ and given a small kick, what will happen?

Accepted Answer

The particle will move away from $$x_3$$; this is unstable equilibrium.

Question 10

Which condition determines whether an equilibrium position is stable or unstable for a potential energy function $$U(s)$$?

Accepted Answer

Stable if $$\left. \frac{d^2$U}{ds^2$} \right|_{s_{eq}} \u003e 0$$, unstable if $$\left. \frac{d^2$U}{ds^2$} \right|_{s_{eq}} \u003c 0$$

Question 11

Which of the following is a correct statement about the total mechanical energy $$E_{mech} in a $$system with only conservative forces acting?

Accepted Answer

$$E_{mech}$$ remains constant over time.

Question 12

A block attached to a spring oscillates back and forth. At the equilibrium position, which of the following is true about its kinetic and potential energies?

Accepted Answer

Kinetic energy is maximum, potential energy is minimum.

Question 13

Which equation correctly represents the energy conservation principle for a system including nonconservative forces and external work?

Accepted Answer

$$K_i + U_i + w_{ext} = K_f + U_f + \Delta E_{th}$$

Question 14

A skier slides down a frictionless hill. Which of the following best describes the transformation of energy as the skier descends?

Accepted Answer

Potential energy decreases, kinetic energy increases, total mechanical energy remains constant.

Question 15

If a force is nonconservative, which of the following statements is true regarding the work done by the force?

Accepted Answer

The work done depends on the path taken between two points.

Question 16

A ball is thrown upward. At its highest point, which of the following is true about its energy?

Accepted Answer

Kinetic energy is zero, potential energy is maximum.

Question 17

Which of the following is a necessary condition for a force to have an associated potential energy?

Accepted Answer

The force must be conservative.

Question 18

A mass attached to a spring is displaced from equilibrium and released. Which of the following best describes the energy changes during one complete oscillation (ignoring friction)?

Accepted Answer

Energy continuously transforms between kinetic and potential, but total mechanical energy remains constant.

Question 19

Which of the following real-world scenarios best illustrates the concept of unstable equilibrium?

Accepted Answer

A pencil balanced perfectly on its tip.

Potential Energy, Conservative Forces, and Energy Diagrams

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