Q: A block slides down a frictionless globe and leaves the surface at angle $$\theta_c$$ from the vertical. What is the relationship between $$\cos\theta_c$$ and $$h_{exit}$$?

$$h_{exit} = R\cos\theta_c + R$$

Question 1

Which of the following correctly expresses the conservation of energy for a system with no work done and no friction present?

Accepted Answer

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

Question 2

A rollercoaster cart starts from rest at height $$h$$ and enters a vertical loop of radius $$R. $$What is the minimum height $$h_{min}$$ required for the cart to safely complete the loop (neglecting friction)?

Accepted Answer

$$h_{min} = \frac{5}{2}R$$

Question 3

In the context of energy conservation, what does $$\Delta E_{th}$$ represent?

Accepted Answer

The change in thermal energy due to dissipative forces like friction

Question 4

A block of mass $$m$$ slides down a frictionless globe of radius $$R$$ and leaves the surface at height $$h_{exit}$$ above the bottom. What is $$h_{exit}$$?

Accepted Answer

$$h_{exit} = \frac{2}{3}R$$

Question 5

Which equation gives the gravitational potential energy of a mass $$m at $$height $$y$$ above the ground?

Accepted Answer

$$U_g = mgy$$

Question 6

For a mass-spring system, what is the formula for elastic potential energy stored in the spring when stretched by $$\Delta x$$ from equilibrium?

Accepted Answer

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

Question 7

If friction is present in a system, how is the change in thermal energy $$\Delta E_{th}$$ calculated?

Accepted Answer

$$\Delta E_{th} = f \Delta r$$

Question 8

At the top of a vertical loop, what is the minimum speed $$v_{min}$$ required for a cart of mass $$m to $$stay on the track?

Accepted Answer

$$v_{min} = \sqrt{gR}$$

Question 9

In the globe problem, what physical condition determines when the block leaves the surface?

Accepted Answer

The normal force $$n$$ becomes zero

Question 10

Which of the following best describes an energy diagram for a mass-spring system?

Accepted Answer

A graph showing the system's potential and total energy as a function of position

Question 11

A mass $$m$$ attached to a spring with spring constant $$k$$ oscillates with amplitude $$A. $$What is the total mechanical energy of the system?

Accepted Answer

$$E = \frac{1}{2}kA^2$

Question 12

In the absence of dissipative forces, what happens to the total mechanical energy of a system?

Accepted Answer

It remains constant

Question 13

A block slides down a frictionless globe and leaves the surface at angle $$	heta_c$$ from the vertical. What is the relationship between $$\cos	heta_c$$ and $$h_{exit}$$?

Accepted Answer

$$h_{exit} = R\cos	heta_c + R$$

Question 14

Which of the following is NOT a correct application of the work-energy principle?

Accepted Answer

Ignoring dissipative forces when friction is present

Question 15

A mass-spring system is displaced from equilibrium by $$x. $$What is the expression for the total energy at this position?

Accepted Answer

$$E = \frac{1}{2}k(x-x_{eq})^2$ + K$$

Question 16

In the rollercoaster loop problem, why is the normal force set to zero at the top of the loop when calculating the minimum speed?

Accepted Answer

Because the cart is on the verge of losing contact with the track

Question 17

Which of the following statements about energy diagrams is correct?

Accepted Answer

If $$\Delta E_{th} = 0$$, the total mechanical energy is constant

Question 18

A block of mass $$m$$ slides down a frictionless globe and leaves the surface at height $$h_{exit}. $$Which conservation law is used to relate its initial and final energies?

Accepted Answer

Conservation of mechanical energy

Question 19

In a mass-spring system, what does the area under the energy diagram curve between two positions represent?

Accepted Answer

The change in potential energy between those positions

Conservation of Energy and Applications in Mechanics

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