Question 1

A block with mass $$m = 20\ \mathrm{kg}$$ slides down a $$60^\circ$$ incline with a coefficient of kinetic friction $$\mu_k = 0.5$$ for a distance $$L = 10\ \mathrm{m}$$ before compressing a spring with spring constant $$k = 200\ \mathrm{N/m}. $$What is the velocity of the block when it first reaches the spring?

Accepted Answer

$$v = \sqrt{2gL(\sin 60^\circ - \mu_k \cos 60^\circ)}$$

Question 2

For the same block and spring system, how much will the spring be compressed when the block comes to rest for the first time?

Accepted Answer

The positive root of $$kx^2 - 2$$mgx(\sin 60^\circ - \mu_k \cos 60^\circ) - 2mgL(\sin 60^\circ - \mu_k \cos 60^\circ) = 0$$

Question 3

What is the total work done by friction from the start until the block comes to rest at maximum spring compression?

Accepted Answer

$$W_f = -\mu_k mg \cos 60^\circ (L + x)$$

Question 4

After the block compresses the spring and is pushed back, what is its velocity as it leaves the spring (assuming it disengages at $$x = 0$$)?

Accepted Answer

$$v = \sqrt{2xg(\sin 60^\circ - \mu_k \cos 60^\circ)}$$

Question 5

A car of mass $$m_1$$ and speed $$v_1$$ collides at a $$30^\circ$$ intersection with a truck of mass $$m_2$$ and speed $$v_2. $$After the collision, the car moves along the truck's original direction and vice versa. If $$m_2 = 2m_1 = 2000$$\ \mathrm{kg}$$ and v_1$ = 3v_2 = 60\ \mathrm{km/h}$$, what percent of the initial kinetic energy is lost to deformation and heat?

Accepted Answer

$$23\%$$

Question 6

A stunt actor and bike (total mass $$M = 80\ \mathrm{kg}) $$start from rest and roll down a $$45^\circ$$ incline of length $$L = 20\ \mathrm{m}$$, then up a frictionless quarter pipe of radius $$R = 5\ \mathrm{m}. $$Each wheel is a thin cylinder ($$m = 3\ \mathrm{kg}$$, $$r = 0.5\ \mathrm{m}). $$What is the velocity of the center of mass as the actor leaves the pipe?

Accepted Answer

$$v_{cm} = \sqrt{\frac{2(M + 2m)gL \sin 45^\circ}{M + 2m}} ($$with $$m$$ for each wheel's rotation)

Question 7

A puck slides on a horizontal surface and comes to rest after $$12\ \mathrm{s}$$ due to friction. How does the energy lost in the first $$6\ \mathrm{s}$$ compare to the energy lost in the last $$6\ \mathrm{s}$$?

Accepted Answer

The energy lost in the first half is greater than in the second half.

Question 8

An explosion splits an object at rest into two pieces of unequal mass. Which piece moves at greater speed?

Accepted Answer

The less massive piece.

Question 9

A ladybug sits at the edge of a merry-go-round that is turning and slowing down. The tangential component of the ladybug's (linear) acceleration is:

Accepted Answer

In the $$-y$$ direction.

Question 10

Compared to the energy required to accelerate a car from rest to $$10\ \mathrm{mph}$$, the energy required to accelerate the same car from $$10\ \mathrm{mph} to$$ $$20\ \mathrm{mph} is$$:

Accepted Answer

Three times as much.

Question 11

A ball is thrown against a wall and bounces back with the same speed. Which statement is true about the ball's kinetic energy and momentum?

Accepted Answer

The kinetic energy of the ball is the same.

Question 12

A ladybug sits at the edge of a merry-go-round that is turning and slowing down. At the instant shown, the radial component of the ladybug's (linear) acceleration is:

Accepted Answer

In the $$-x$$ direction.

Question 13

A block of mass $$M$$ moves in a circle on a frictionless table, attached to a string pulled through a hole at the center (see image). As the string is pulled, which of the following is true?

Accepted Answer

The angular momentum of $$M$$ remains constant.

Question 14

A uniform disk of radius $$R$$ and mass $$M$$ spins with angular velocity $$\omega$$ about an axis through its center. What is its angular momentum if it spins about a parallel axis a distance $$h$$ away?

Accepted Answer

$$(I + Mh^2$)\omega$$

Question 15

An ice skater with rotational inertia $$I_0$$ spins with angular speed $$\$$omega_0$$. She pulls her arms in, increasing her angular speed to 4$$\$$omega_0. $$What is her new rotational inertia?

Accepted Answer

$$I_0/4$

Question 16

The angular momentum vector of Earth, due to its daily rotation, is directed:

Accepted Answer

North.

Conservation of Angular Momentum for a Block on a Table

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