Q: A 400 kg roller coaster cart starts at rest at point A, which is 125 m above the ground, and enters a frictionless loop-the-loop with a diameter of 50 m. What is the speed of the cart at the top of the loop (point B)? (Assume $$g = 9.8 \ \mathrm{m/s^2$}$$ and neglect air resistance.)

$$v = \sqrt{2g(h - r)} = \sqrt{2 \times 9.8 \times (125 - 25)} = 42.8 \ \mathrm{m/s}$$

Question 1

A 400 kg roller coaster cart starts at rest at point A, which is 125 m above the ground, and enters a frictionless loop-the-loop with a diameter of 50 m. What is the speed of the cart at the top of the loop (point B)? (Assume $$g = 9.8 \ \mathrm{m/s^2$}$$ and neglect air resistance.)

Accepted Answer

$$v = \sqrt{2g(h - r)} = \sqrt{2 	imes 9.8 	imes (125 - 25)} = 42.8 \ \mathrm{m/s}$$

Question 2

At point B (top of the loop), what is the normal force exerted on the 400 kg cart? (Loop diameter is 50 m, $$g = 9.8 \ \mathrm{m/s^2$})$$

Accepted Answer

$$F_N = m \left( \frac{v^2$}{r} - g \right ) = 400 \left( \frac{(42.8)^2$}{25} - 9.8 \right ) = 19,550 \ \mathrm{N}$$

Question 3

If the cart travels 200 m over a rough set of rails after the loop and comes to a stop, what is the coefficient of friction $$\mu$$? (Initial height $$h = 125 m$$, $$d = 200 m$$, $$g = 9.8 \ \mathrm{m/s^2$})$$

Accepted Answer

$$\mu = \frac{h}{d} = \frac{125}{200} = 0.625$$

Question 4

Which principle allows you to solve for the coefficient of friction in the second part of the problem without using kinematics?

Accepted Answer

Conservation of energy

Question 5

If the loop-the-loop had friction, which of the following would be true about the cart's speed at point B?

Accepted Answer

The speed at point B would be less than in the frictionless case.

Question 6

Which of the following best describes the energy transformation as the cart moves from point A to point B?

Accepted Answer

Potential energy is converted to kinetic energy.

Question 7

If the cart had started with an initial speed at point A, how would this affect the normal force at point B?

Accepted Answer

The normal force at point B would increase.

Question 8

Which equation correctly relates the work done by friction to the loss in mechanical energy as the cart comes to a stop over the rough rails?

Accepted Answer

$$W_{friction} = \mu m g d$$

Question 9

If the diameter of the loop is doubled, what happens to the required speed at the top of the loop for the cart to stay on the track?

Accepted Answer

The required speed increases.

Question 10

Which of the following is a correct application of Newton's second law at the top of the loop?

Accepted Answer

$$F_N + mg = m \frac{v^2$}{r}$$

Question 11

If the cart's mass is doubled, what happens to the coefficient of friction required to stop the cart over the same distance?

Accepted Answer

The coefficient of friction remains the same.

Question 12

Which of the following is the correct expression for the cart's kinetic energy at point B?

Accepted Answer

$$KE_B = m g (h - r)$$

Question 13

If the cart does not have enough speed at the top of the loop, what is most likely to happen?

Accepted Answer

The cart will fall off the track.

Question 14

Which of the following best explains why the normal force at the top of the loop can be zero?

Accepted Answer

The gravitational force alone provides the required centripetal force.

Question 15

If the cart starts at a lower height, which of the following is true about its ability to complete the loop?

Accepted Answer

It may not have enough energy to complete the loop.

Question 16

Which of the following is the correct formula for the minimum speed required at the top of the loop for the cart to stay on the track?

Accepted Answer

$$v_{min} = \sqrt{g r}$$

Loop-the-Loop Dynamics and Energy Conservation in Roller Coaster Physics

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