Q: At the bottom of the quarter-circle ramp, what is the normal force acting on the skateboarder (mass 25.0 kg, radius $$R = 3.00 m$$, speed $$v = 7.67 m/s$$)?

$$n = m\left(\frac{v^2$}{R} + g\right) = 25.0 \left(\frac{(7.67)^2$}{3.00} + 9.8\right) = 25.0 \times (19.6 + 9.8) = 25.0 \times 29.4 = 735 N$$

Q: A 2000 kg elevator falls at 8.00 m/s and compresses a spring by 3.00 m. What is the required spring constant $$k to $$stop the elevator, ignoring friction?

$$k = \frac{2(K_i + U_{grav})}{x^2$} = \frac{2(\frac{1}{2} \times 2000 \times 8^2$ + 2000 \times 9.81 \times 3)}{(3)^2$} = 2.37 \times 10^4$ N/m

Q: A crate slides up a ramp and then back down. If the friction force is 23 N and the crate travels 1.6 m up and 1.6 m down, what is the total work done by friction?

$$W_{friction} = -f \times (1.6 + 1.6) = -23 \times 3.2 = -74 J$$

Q: If you lift a 10 kg box 2 m in 4 seconds, what is your average power output?

$$P = \frac{W}{t} = \frac{mgh}{t} = \frac{10 \times 9.8 \times 2}{4} = 49 W$$

Question 1

A 25.0 kg skateboarder starts from rest at the top of a frictionless quarter-circle ramp with radius $$R = 3.00 m. $$What is his speed at the bottom of the ramp?

Accepted Answer

$$v = \sqrt{2gR} = 7.67 m/s$$

Question 2

At the bottom of the quarter-circle ramp, what is the normal force acting on the skateboarder (mass 25.0 kg, radius $$R = 3.00 m$$, speed $$v = 7.67 m/s$$)?

Accepted Answer

$$n = m\left(\frac{v^2$}{R} + gight) = 25.0 \left(\frac{(7.67)^2$}{3.00} + 9.8ight) = 25.0 	imes (19.6 + 9.8) = 25.0 	imes 29.4 = 735 N$$

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 2000 kg elevator falls at 8.00 m/s and compresses a spring by 3.00 m. What is the required spring constant $$k to $$stop the elevator, ignoring friction?

Accepted Answer

$$k = \frac{2(K_i + U_{grav})}{x^2$} = \frac{2(\frac{1}{2} 	imes 2000 	imes 8^2$ + 2000 	imes 9.81 	imes 3)}{(3)^2$} = 2.37 	imes 10^4$ N/m

Question 5

A block slides down a frictionless ramp from height $$y_1$$ and another block slides down a straight ramp from the same height. Which block arrives at the bottom with greater speed?

Accepted Answer

Both blocks arrive at the bottom with the same speed.

Question 6

A crate of mass 8.0 kg is given an initial speed of 5.0 m/s up a 2.5 m ramp inclined at $$30^	ext{o}$$, but friction causes it to stop after 1.6 m. What is the magnitude of the friction force?

Accepted Answer

$$f = \frac{K_i - U_f}{s} = \frac{100 - 62.7}{1.6} = 23 N$$

Question 7

If a block compresses a spring while sliding down a frictionless incline, what happens to its gravitational and elastic potential energies?

Accepted Answer

Gravitational potential energy decreases; elastic potential energy increases.

Question 8

Which of the following is true about nonconservative forces?

Accepted Answer

Nonconservative forces convert mechanical energy into other forms, such as heat.

Question 9

A block of mass $$m is $$released from rest at height $$h on a $$frictionless ramp. What is its speed at the bottom?

Accepted Answer

$$v = \sqrt{2gh}$$

Question 10

A crate slides up a ramp and then back down. If the friction force is 23 N and the crate travels 1.6 m up and 1.6 m down, what is the total work done by friction?

Accepted Answer

$$W_{friction} = -f 	imes (1.6 + 1.6) = -23 	imes 3.2 = -74 J$$

Question 11

What is the formula for the elastic potential energy stored in a spring compressed by $$x$$ meters with spring constant $$k$$?

Accepted Answer

$$U_{el} = \frac{1}{2} k x^2$

Question 12

A runner travels up three different paths to the summit of a mountain. Which path requires the most energy, assuming only gravity acts?

Accepted Answer

All paths require the same energy because gravity is a conservative force.

Question 13

What is the SI unit of power?

Accepted Answer

Watt (W)

Question 14

If you lift a 10 kg box 2 m in 4 seconds, what is your average power output?

Accepted Answer

$$P = \frac{W}{t} = \frac{mgh}{t} = \frac{10 	imes 9.8 	imes 2}{4} = 49 W$$

Question 15

Which equation correctly expresses instantaneous power for a force acting on a particle?

Accepted Answer

$$p = \vec{F} \cdot \vec{v}$$

Question 16

Two blocks are connected by a rope and pulley. When released, block $$m_1$$ slides down a ramp and block $$m_2$$ moves upward. After each block has moved a distance $$d$$, how does the total work done on $$m_1$$ compare to $$m_2$$?

Accepted Answer

The total work done on $$m_1 is $$the same as the total work done on $$m_2.$$

Guided Study: Energy Conservation and Forces on a Curved Ramp

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