Q: A spring with spring constant $$k is $$compressed a length $$z$$ and a mass $$m is $$placed at its free end on a frictionless slope which makes an angle $$\theta$$ with respect to the horizontal. The spring is released from rest. If the mass is not attached to the spring, how far up the slope will the mass reach?

$$d = \frac{kz^2$}{2mg\sin\theta}$$

Q: A pendulum is formed from a small ball of mass $$m on a $$string of length $$L. A $$peg is height $$h = \frac{L}{3}$$ above the pendulum's lowest point. From what minimum angle must the pendulum be released in order for the ball to go over the top of the peg without the string going slack?

$$\theta_{min} = \arccos\left(\frac{2}{3}\right)$$

Question 1

A bowling ball and a ping-pong ball have the same momentum. Which has the greater kinetic energy?

Accepted Answer

The ping-pong ball

Question 2

A little girl uses a slingshot to launch a pebble straight up into the air with an initial speed $$v_0. $$Using conservation of energy, how high does the pebble go?

Accepted Answer

$$h = \frac{v_0$^2}{2g}$$

Question 3

Ball A has half the mass and eight times the kinetic energy of Ball B. What is the speed ratio $$v_A/v_B$$?

Accepted Answer

4

Question 4

On a frictionless table, a mass $$m$$ moving at speed $$v$$ collides with another mass $$m$$ initially at rest. The masses stick together. How much energy is converted to heat?

Accepted Answer

$$\frac{1}{4}mv^2$

Question 5

A paddle hits a ping-pong ball. The paddle moves with speed $$v to $$the right. The ping-pong ball moves with speed $$v to $$the left. What is the resulting speed of the ping-pong ball? (Assume the paddle is much more massive than the ping-pong ball.)

Accepted Answer

$$2v$$

Question 6

An object slides down a steeper hill and a less steep hill from the same height. Both hills are frictionless. For which is it going faster at the bottom?

Accepted Answer

Same speed at the bottom of both hills

Question 7

A roller-coaster car of mass $$m$$ starts from rest at height $$h$$ and goes through a loop of radius $$R. $$What is the minimum height $$h$$ such that the car makes it around the loop?

Accepted Answer

$$h = \frac{5}{2}R$$

Question 8

A spring with spring constant $$k is $$compressed a length $$z$$ and a mass $$m is $$placed at its free end on a frictionless slope which makes an angle $$	heta$$ with respect to the horizontal. The spring is released from rest. If the mass is not attached to the spring, how far up the slope will the mass reach?

Accepted Answer

$$d = \frac{kz^2$}{2mg\sin	heta}$$

Question 9

A 1 × $$10^3 W $$hair dryer is used for 60 s or a 10 W night light left on for 24 h. Which consumes more energy?

Accepted Answer

The night light

Question 10

A 5.0 kg cat leaps from the floor to the top of a 1 m-high table. If the cat pushes against the floor for 0.2 s to accomplish this feat, what was her average power output during the push-off period?

Accepted Answer

$$P = 250 W$$

Question 11

A red sphere of mass $$m$$ and a blue sphere of mass $$2m$$ are attached to the ceiling by massless strings of identical length forming two pendulums of length $$L. $$The red sphere is drawn to the left so that its center of mass has been raised a distance $$h$$ and is then released. As the red sphere moves back through equilibrium it collides with the blue sphere head-on. If the two spheres adhere together during the collision, what is the final height $$h_f$$?

Accepted Answer

$$h_f = \frac{h}{3}$$

Question 12

Tarzan, with mass $$m = 70.2 kg$$, swings from a cliff at the end of a vine $$L = 18 m $$long. From the top of the cliff to the lowest point, the swing descends by $$\Delta y = 7.2 m. $$The vine will break if the force on it exceeds $$T_{max} = 950 N. $$Does the vine break?

Accepted Answer

No, the vine does not break

Question 13

A block with mass $$m$$ slides with speed $$v$$ along a frictionless table toward a stationary block that also has mass $$m. A $$massless spring with spring constant $$k is $$attached to the second block. What is the maximum distance the spring gets compressed?

Accepted Answer

$$x_{max} = v\sqrt{\frac{2m}{k}}$$

Question 14

A pendulum is formed from a small ball of mass $$m on a $$string of length $$L. A $$peg is height $$h = \frac{L}{3}$$ above the pendulum's lowest point. From what minimum angle must the pendulum be released in order for the ball to go over the top of the peg without the string going slack?

Accepted Answer

$$	heta_{min} = \arccos\left(\frac{2}{3}ight)$$

Question 15

Which of the following statements about energy conservation in isolated systems is correct?

Accepted Answer

The total energy, including kinetic and potential energies, remains constant over time.

Question 16

A mass $$m is $$attached to a spring with spring constant $$k$$ and is stretched (or compressed) by a distance $$x$$ from its equilibrium length. What is the elastic potential energy stored in the spring?

Accepted Answer

$$U_{elastic} = \frac{1}{2}kx^2$

Question 17

A cart of mass $$m$$ travels vertically down a track with frictionless wheels, initially at rest at height $$h$$ above the ground. When the cart reaches the bottom, what is its speed?

Accepted Answer

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

Question 18

Which formula correctly expresses the kinetic energy $$K of an $$object with mass $$m$$ and speed $$v$$?

Accepted Answer

$$K = \frac{1}{2}mv^2$

Conservation of Energy and Applications in Mechanics

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