Question 1

A particle moves in a uniform gravitational field near the surface of a planet. What is the change in potential energy as it moves from height $$y_0 to$$ $$y_1$$?

Accepted Answer

$$U(y_1$) - U(y_0$) = m g (y_1$ - y_0$)$$

Question 2

A spring obeys Hooke's law with force $$\vec{F}_s = -k x \hat{i}. $$What is the change in potential energy as the spring is stretched from $$x_0 to$$ $$x_1$$?

Accepted Answer

$$U(x_1$) - U(x_0$) = \frac{1}{2} k (x_1$^2 - x_0$^2)$$

Question 3

The gravitational force between two masses $$m_1$$ and $$m_2$$ separated by distance $$r is$$ $$\vec{F}_G = -G \frac{m_1$ m_2$}{r^2$} \hat{r}. $$What is the change in potential energy as the separation changes from $$r_0 to$$ $$r_1$$?

Accepted Answer

$$U(r_1$) - U(r_0$) = G m_1$ m_2$ \left( \frac{1}{r_1$} - \frac{1}{r_0$} \right)$$

Question 4

A ball of mass $$m is $$attached to a string of length $$L$$ and swings in a vertical circle. What is the tension $$T_b in $$the string at the bottom of the circle if the ball's speed there is $$v_b$$?

Accepted Answer

$$T_b = m \left( g + \frac{v_b^2$}{L} \right)$$

Question 5

A small bird flies at an average speed of $$10.7\ \mathrm{m/s}$$ and consumes $$3.70\ \mathrm{W} of $$power. If it uses $$4.00\ \mathrm{g} of $$fat (energy density $$9.40\ \mathrm{Cal/g}$$, $$1\ \mathrm{Cal} = 4184\ \mathrm{J}$$), how far can it fly before needing to feed again?

Accepted Answer

$$d = 455\ \mathrm{km}$$

Question 6

A $$1.65\ \mathrm{m}$$ tall person lifts a $$1.55\ \mathrm{kg}$$ book to a height of $$2.05\ \mathrm{m}$$ above the ground. What is the potential energy of the book relative to the ground?

Accepted Answer

$$U_{\mathrm{ground}} = 31.1\ \mathrm{J}$$

Question 7

A ski starts from rest and slides down a $$21^{\circ}$$ incline $$85\ \mathrm{m}$$ long. If the coefficient of friction is $$0.075$$, what is the ski's speed at the base of the incline?

Accepted Answer

$$v = 22\ \mathrm{m/s}$$

Question 8

A $$51\ \mathrm{kg}$$ student runs at $$4.5\ \mathrm{m/s}$$, grabs a hanging rope, and swings out over a lake. He releases the rope when his velocity is zero. What is the angle $$	heta$$ when he releases the rope?

Accepted Answer

$$	heta = 26$$^{\circ}$$

Question 9

Which of the following expressions gives the distance $$d$$ traveled in the time interval $$t at an $$average speed $$v$$?

Accepted Answer

$$d = v t$$

Question 10

A ball of mass $$m is $$swung in a vertical circle of radius $$L. $$What is the tension $$T_t in $$the string at the top of the circle if the ball's speed there is $$v_t$$?

Accepted Answer

$$T_t = m \left( \frac{v_t^2$}{L} - g \right)$$

Question 11

A migrating hummingbird flies nonstop across the Gulf of Mexico ($$800\ \mathrm{km}) at$$ $$40.0\ \mathrm{km/h}$$, consuming energy at $$1.70\ \mathrm{W}. $$How much fat (in grams, $$9.40\ \mathrm{Cal/g}$$, $$1\ \mathrm{Cal} = 4184\ \mathrm{J}) $$does it need for the flight?

Accepted Answer

$$m_{\mathrm{fat}} = 3.11\ \mathrm{g}$$

Question 12

Which equation correctly relates the work $$W$$ done by gravity when lifting a book from the ground to the top of a person's head?

Accepted Answer

$$W = U_{\mathrm{head}} - U_{\mathrm{ground}}$$

Question 13

A skier slides down a $$21^{\circ}$$ incline $$85\ \mathrm{m}$$ long, then continues on level snow with the same coefficient of friction ($$0.075). $$How far will the ski travel along the level before stopping?

Accepted Answer

$$d = 330\ \mathrm{m}$$

Question 14

A ball is swung in a vertical circle. If the speed at the bottom is $$v_b$$ and at the top is $$v_t$$, what is the relationship between them using energy conservation?

Accepted Answer

$$v_b^2$ = v_t^2$ + 4 g L$$

Question 15

A bird consumes energy at a rate of $$3.70\ \mathrm{W}. If it $$has $$4.00\ \mathrm{g} of $$fat available ($$9.40\ \mathrm{Cal/g}$$, $$1\ \mathrm{Cal} = 4184\ \mathrm{J}$$), how many hours can it fly before needing to feed?

Accepted Answer

$$t = 1.18\ \mathrm{hr}$$

Question 16

Which of the following is true for the change in potential energy $$\Delta U in a $$conservative force field?

Accepted Answer

$$\Delta U = - \int_{1}^{2} \vec{F} \cdot d\vec{s}$$

Question 17

A ball of mass $$m is $$swung in a vertical circle of radius $$L. $$What is the magnitude of the net force in the $$y$$ direction acting on the ball at the bottom of the circle?

Accepted Answer

$$T_b - m g$$

Potential Energy, Work, and Applications in Mechanics

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