Q: A box of mass $$m = 10\ \mathrm{kg} is $$pushed up a frictionless ramp inclined at $$30^\circ to $$the horizontal by a constant force $$F = 100\ \mathrm{N}$$ parallel to the ramp. If the box moves a distance $$d = 5\ \mathrm{m} up $$the ramp, what is the work done by the force $$F$$?

$$W = Fd \cos \theta = 100 \times 5 \times \cos 30^\circ = 433\ \mathrm{J}$$

Q: A $$2\ \mathrm{kg}$$ block is released from rest at the top of a frictionless ramp of height $$h = 3\ \mathrm{m}. $$What is its speed at the bottom of the ramp?

$$v = \sqrt{2gh} = \sqrt{2 \times 9.8 \times 3} = 7.67\ \mathrm{m/s}$$

Q: A $$0.5\ \mathrm{kg}$$ mass is attached to a spring with spring constant $$k = 200\ \mathrm{N/m}$$ and stretched $$x = 0.1\ \mathrm{m}$$ from equilibrium. What is the potential energy stored in the spring?

$$U = \frac{1}{2}kx^2$ = \frac{1}{2} \times 200 \times (0.1)^2$ = 1\ \mathrm{J}$$

Q: A $$60\ \mathrm{kg}$$ woman runs up a $$3.0\ \mathrm{m}$$ high flight of stairs in $$4.0\ \mathrm{s}$$, reaching a final speed of $$2.0\ \mathrm{m/s}. $$What is her average power output?

$$P = \frac{mgh + \frac{1}{2}mv^2$}{t} = \frac{60 \times 9.8 \times 3 + 0.5 \times 60 \times (2)^2$}{4} = 480\ \mathrm{W}$$

Q: A $$1.6\ \mathrm{kg}$$ elevator is lifted at constant speed by a motor. The elevator and passengers have a combined mass of $$200\ \mathrm{kg}$$, and a constant frictional force of $$4000\ \mathrm{N}$$ retards its motion. What power must the motor deliver to lift the elevator at $$2.0\ \mathrm{m/s}$$?

$$P = (mg + F_{\mathrm{friction}})v = (200 \times 9.8 + 4000) \times 2.0 = 7,920\ \mathrm{W}$$

Q: A $$0.2\ \mathrm{kg}$$ block is compressed against a spring with $$k = 50\ \mathrm{N/m} by$$ $$x = 0.1\ \mathrm{m}$$ and released on a frictionless surface. What is its speed when it leaves the spring?

$$v = \sqrt{\frac{kx^2$}{m}} = \sqrt{\frac{50 \times (0.1)^2$}{0.2}} = 1.58\ \mathrm{m/s}$$

Question 1

A box of mass $$m = 10\ \mathrm{kg} is $$pushed up a frictionless ramp inclined at $$30^\circ to $$the horizontal by a constant force $$F = 100\ \mathrm{N}$$ parallel to the ramp. If the box moves a distance $$d = 5\ \mathrm{m} up $$the ramp, what is the work done by the force $$F$$?

Accepted Answer

$$W = Fd \cos 	heta = 100 	imes 5 	imes \cos 30^\circ = 433\ \mathrm{J}$$

Question 2

Which of the following statements about kinetic energy is correct?

Accepted Answer

Kinetic energy is a scalar quantity and is always positive for a moving object.

Question 3

A $$2\ \mathrm{kg}$$ block is released from rest at the top of a frictionless ramp of height $$h = 3\ \mathrm{m}. $$What is its speed at the bottom of the ramp?

Accepted Answer

$$v = \sqrt{2gh} = \sqrt{2 	imes 9.8 	imes 3} = 7.67\ \mathrm{m/s}$$

Question 4

A wind turbine converts wind energy into electrical energy. Which forms of energy are involved in this process?

Accepted Answer

Kinetic energy of wind is converted into mechanical energy of the turbine, then into electrical energy.

Question 5

A $$0.5\ \mathrm{kg}$$ mass is attached to a spring with spring constant $$k = 200\ \mathrm{N/m}$$ and stretched $$x = 0.1\ \mathrm{m}$$ from equilibrium. What is the potential energy stored in the spring?

Accepted Answer

$$U = \frac{1}{2}kx^2$ = \frac{1}{2} 	imes 200 	imes (0.1)^2$ = 1\ \mathrm{J}$$

Question 6

Which of the following is a non-conservative force?

Accepted Answer

Friction

Question 7

A $$60\ \mathrm{kg}$$ woman runs up a $$3.0\ \mathrm{m}$$ high flight of stairs in $$4.0\ \mathrm{s}$$, reaching a final speed of $$2.0\ \mathrm{m/s}. $$What is her average power output?

Accepted Answer

$$P = \frac{mgh + \frac{1}{2}mv^2$}{t} = \frac{60 	imes 9.8 	imes 3 + 0.5 	imes 60 	imes (2)^2$}{4} = 480\ \mathrm{W}$$

Question 8

A $$1.6\ \mathrm{kg}$$ elevator is lifted at constant speed by a motor. The elevator and passengers have a combined mass of $$200\ \mathrm{kg}$$, and a constant frictional force of $$4000\ \mathrm{N}$$ retards its motion. What power must the motor deliver to lift the elevator at $$2.0\ \mathrm{m/s}$$?

Accepted Answer

$$P = (mg + F_{\mathrm{friction}})v = (200 	imes 9.8 + 4000) 	imes 2.0 = 7,920\ \mathrm{W}$$

Question 9

If only conservative forces act on a system, which of the following statements is true about the total mechanical energy $$E$$?

Accepted Answer

Total mechanical energy $$E = K + U is $$conserved throughout the motion.

Question 10

A $$0.2\ \mathrm{kg}$$ block is compressed against a spring with $$k = 50\ \mathrm{N/m} by$$ $$x = 0.1\ \mathrm{m}$$ and released on a frictionless surface. What is its speed when it leaves the spring?

Accepted Answer

$$v = \sqrt{\frac{kx^2$}{m}} = \sqrt{\frac{50 	imes (0.1)^2$}{0.2}} = 1.58\ \mathrm{m/s}$$

Question 11

Which of the following best describes the work-energy theorem?

Accepted Answer

The net work done on an object equals the change in its kinetic energy: $$W_{\mathrm{net}} = K_f - K_i$$

Question 12

A $$5\ \mathrm{kg}$$ object is dropped from a height of $$10\ \mathrm{m}. $$Ignoring air resistance, what is its kinetic energy just before it hits the ground?

Accepted Answer

$$K = mgh = 5 	imes 9.8 	imes 10 = 490\ \mathrm{J}$$

Question 13

A force $$F$$ acts on an object, causing a displacement $$s at an $$angle $$	heta to $$the direction of the force. What is the general formula for the work done by the force?

Accepted Answer

$$W = F s \cos 	heta$$

Question 14

A $$3\ \mathrm{kg}$$ cart starts from rest and is pushed by a constant force along a frictionless surface for $$4\ \mathrm{m}$$, reaching a speed of $$8\ \mathrm{m/s}. $$What is the net work done on the cart?

Accepted Answer

$$W = \Delta K = \frac{1}{2} m v^2$ - 0 = 0.5 	imes 3 	imes 8^2$ = 96\ \mathrm{J}$$

Question 15

Which of the following is true about the work done by gravity when an object moves from height $$y_i to$$ $$y_f$$?

Accepted Answer

$$W_{\mathrm{gravity}} = mg(y_i - y_f)$$

Question 16

A $$0.3\ \mathrm{kg}$$ mass is attached to a spring with $$k = 120\ \mathrm{N/m}$$ and compressed $$0.05\ \mathrm{m}$$ from equilibrium. If released, what is its speed at equilibrium?

Accepted Answer

$$v = \sqrt{\frac{kx^2$}{m}} = \sqrt{\frac{120 	imes (0.05)^2$}{0.3}} = 1.0\ \mathrm{m/s}$$

Question 17

A leaf falls to the ground at constant speed due to air resistance. How does its final mechanical energy compare to its initial mechanical energy?

Accepted Answer

Final mechanical energy is less than initial mechanical energy due to work done by non-conservative air resistance.

Chapter 7: Work and Energy – Physics Study Notes

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