Q: If the incline in the previous problem is free to move (no friction anywhere), what is the acceleration of the box relative to the ground? (Express as a vector in components.)

$$\vec{a} = \left( \frac{m \sin \theta \cos \theta}{m + M} g, \; g \sin^2$ \theta \frac{M}{m + M} \right)$$

Q: What is the minimum coefficient of static friction $$\mu_s$$ between the incline and the table so that the incline does not move when the box slides down?

$$\mu_s = \frac{m}{M} \sin \theta$$

Question 1

A string is fixed at one end and has a mass hanging on the other end. What property of the string ensures uniform tension along its length?

Accepted Answer

The string must be massless, so that the tension is the same at every point along its length.

Question 2

Why do you sometimes get very high friction between two smooth flat surfaces of the same material, such as aluminum on aluminum?

Accepted Answer

Because microscopic welding or cold welding can occur at the contact points, increasing friction.

Question 3

When a washing machine spins clothes to remove water, is it correct to say that water is removed by centrifugal force?

Accepted Answer

No, the water moves outward due to inertia, not a real force; the drum pushes the clothes inward, but not the water.

Question 4

A volcanic bomb is ejected from point A at the top of a volcano with initial speed $$v_0 at an $$angle $$\$$theta_0$$ above the horizontal. The bomb lands at point B, a horizontal distance d$ and vertical distance h$ below A. What is the equation for the initial speed v_0$ required to reach B? (Ignore air resistance.)

Accepted Answer

$$v_0$$ = \sqrt{ \frac{g $$d^2$$}{2 \$$cos^2$$ \$$theta_0 (d$$ 	an \$$theta_0 + h$$)} }$$

Question 5

For the volcanic bomb problem, what is the time of flight $$t in $$terms of $$d$$, $$h$$, $$\$$theta_0$$, and v_0$?

Accepted Answer

$$t = \frac{d}{v_0$ \cos \$$theta_0$$}$$

Question 6

A centrifuge rotates a rider at a distance $$R$$ from the center. At what angular velocity $$\omega$$ will the centripetal acceleration equal $$A$$?

Accepted Answer

$$\omega = \sqrt{ \frac{A}{R} }$$

Question 7

In the astronaut centrifuge, the cage hangs on a pivot and swings outward during rotation. At what angle $$	heta$$ below the horizontal will the cage hang when the centripetal acceleration is $$A$$?

Accepted Answer

$$	an 	heta = \frac{A}{g}$$

Question 8

A box of mass $$m is $$placed on an incline of mass $$M at $$angle $$	heta$$ with the horizontal. The incline is fixed. What is the acceleration of the box along the incline (no friction)?

Accepted Answer

$$a = g \sin 	heta$$

Question 9

If the incline in the previous problem is free to move (no friction anywhere), what is the acceleration of the box relative to the ground? (Express as a vector in components.)

Accepted Answer

$$\vec{a} = \left( \frac{m \sin 	heta \cos 	heta}{m + M} g, \; g \sin^2$ 	heta \frac{M}{m + M} ight)$$

Question 10

What is the minimum coefficient of static friction $$\mu_s$$ between the incline and the table so that the incline does not move when the box slides down?

Accepted Answer

$$\mu_s = \frac{m}{M} \sin 	heta$$

Question 11

A student claims that the tension in a heavy rope holding a mass is the same everywhere. Which of the following best explains why this is incorrect?

Accepted Answer

The tension increases toward the fixed end because each segment must support the weight of the rope below it plus the hanging mass.

Question 12

A projectile is launched from a height $$h$$ above the ground with speed $$v_0 at $$angle $$\$$theta_0$$. What is the general equation for the horizontal range d$?

Accepted Answer

$$d = \frac{v_0$ \cos \$$theta_0$$}{g} \left( v_0$ \sin \$$theta_0$$ + \sqrt{ (v_0$ \sin \$$theta_0$$)^2 + 2gh$$ } \right)$

Question 13

In the astronaut centrifuge, if the radius is doubled but the angular velocity is kept constant, what happens to the centripetal acceleration?

Accepted Answer

It quadruples.

Question 14

A box slides down a frictionless incline that is free to move on a frictionless surface. What happens to the center of mass of the box-incline system?

Accepted Answer

It remains stationary because there are no external horizontal forces.

Question 15

If the coefficient of friction between the box and the incline is increased, what happens to the acceleration of the box down the incline?

Accepted Answer

It decreases.

Physics 7A Midterm Study Guidance: Mechanics, Forces, and Motion

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