Q: What is the maximum velocity that a car of mass $$m$$ can have on the banked curve described above to still perform uniform circular motion?

$$v = \sqrt{rg \frac{\mu_s + \tan \theta}{1 - \mu_s \tan \theta}}$$

Question 1

A child on a sled is being pulled to the right with a force $$\vec{F} at an $$angle $$	heta$$ above the horizontal, as shown in the first image. If the sled is speeding up to the right, what is the net force acting on the child/sled system in vector notation? (Let $$\hat{i}$$ and $$\hat{j} be $$unit vectors in the horizontal and vertical directions, respectively, and $$F_{net} be $$the magnitude of the net force.)

Accepted Answer

$$F_{net} = F_{net} \hat{i} + 0 \hat{j}$$

Question 2

Refer to the two images of a person pulling and pushing a sled at the same angle $$	heta$$ and with the same force $$F. In $$which case will the sled experience more friction force, and why?

Accepted Answer

Case 2 will have more friction because by pushing down on the child/sled system the man is increasing the normal force that snow applies on the sled and consequentially increasing friction force.

Question 3

A variable force is exerted on a mass $$m$$ during a time interval $$\Delta t$$, changing its velocity from $$v_0 to$$ $$v_f. $$What is the average force exerted during this time interval?

Accepted Answer

$$F_{average} = m \frac{v_f - v_0$}{\Delta t}$$

Question 4

A 20-N bucket is at the bottom of a 9-m deep well and is initially at rest. The bucket is then pulled to the top of the well by a string that exerts a constant tension force such that it reaches the top in 3.0 s. During this interval, the tension in the string is:

Accepted Answer

24 N

Question 5

A block of mass $$m$$ rests on a table. You apply a small force to the block, but the block does not move. Which of the following is an action/reaction pair according to Newton’s third law?

Accepted Answer

The block’s push on the table and the table’s push on the block

Question 6

In the Atwood machine shown (seventh image), if the masses start motion from rest, which graph best represents the velocity of the smaller mass $$m as a $$function of time? Assume that vertically up is the positive direction.

Accepted Answer

A curved line increasing with time (indicating acceleration)

Question 7

If mass $$M$$ has acceleration $$a_M$$ and mass $$m$$ has acceleration $$a_m in $$the Atwood machine, what is the correct $$(a_M, a_m)$$ pair? (Let $$g be $$the magnitude of gravitational acceleration.)

Accepted Answer

$$(a_M = \frac{g(M-m)}{M+m} \hat{j},\ a_m = \frac{g(M-m)}{M+m} \hat{j})$$

Question 8

Which free body diagram correctly represents the forces acting on mass $$M in $$the Atwood machine?

Accepted Answer

A diagram with $$F_T ($$tension) upward and $$F_g ($$gravity) downward (second image)

Question 9

What is the magnitude of the force that the pin provides to the pulley in the Atwood machine, so the pulley is not moving vertically? (Assume the pulley is massless.)

Accepted Answer

$$F_{pin} = 2(M + m)g$$

Question 10

A car of mass $$m$$ rounds a banked curve of radius $$r$$ and angle $$	heta as $$shown in the fourth image. Which force provides the centripetal acceleration that keeps the car in circular motion?

Accepted Answer

A combination of $$F_f$$ and $$F_N$$

Question 11

What is the maximum velocity that a car of mass $$m$$ can have on the banked curve described above to still perform uniform circular motion?

Accepted Answer

$$v = \sqrt{rg \frac{\mu_s + 	an 	heta}{1 - \mu_s 	an 	heta}}$$

Question 12

If a car hits a patch of dark ice and the coefficient of static friction between the car and ice drops to zero, what will happen to the motion of the car and why?

Accepted Answer

The car will slide off the road, increasing the radius of the path due to the tendency of an object to maintain direction and magnitude of velocity if a force stops affecting it.

Question 13

Refer to the asteroid system shown in the fifth image. A student states: “Asteroid A will attract Asteroid B with more force than Asteroid B will exert on Asteroid A because Asteroid A has bigger mass than Asteroid B.” Do you agree or disagree with the student, and why?

Accepted Answer

Disagree because the statement contradicts Newton’s third law.

Question 14

For the asteroid system shown, find the net force each asteroid experiences and choose the correct ranking of magnitudes $$||F_{net}||.$$

Accepted Answer

$$||F_{net}||_A = ||F_{net}||_C \u003e ||F_{net}||_B$$

Question 15

If asteroids B and C are orbiting asteroid A with uniform circular motion, what is the correct relationship of the periods for the two asteroids?

Accepted Answer

$$T_C = 4 T_B$$

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