Q: Two masses, $$m_1$$ and $$m_2$$, are connected by a light string over a frictionless pulley. $$m_1 = 2.00$$\ \mathrm{kg}$$ rests on a frictionless incline at $$\theta = 55.0^\circ$$, and m_2$ = 6.00\ \mathrm{kg}$$ hangs vertically. What is the correct expression for the acceleration $$a of $$the system?

$$a = \dfrac{m_2$ g - m_1$ g \sin \theta}{m_1$ + m_2$}$$

Q: For the same system, what is the tension $$T in $$the string connecting $$m_1$$ and $$m_2$$?

$$T = m_1$ (g \sin \theta + a)$$

Q: Suppose the incline is made steeper in the first system. What happens to the acceleration $$a$$?

$$a$$ increases because $$m_1 g$$ \sin \theta$$ increases, reducing the net force opposing m_2$.

Question 1

Two masses, $$m_1$$ and $$m_2$$, are connected by a light string over a frictionless pulley. $$m_1 = 2.00$$\ \mathrm{kg}$$ rests on a frictionless incline at $$	heta = 55.0^\circ$$, and m_2$ = 6.00\ \mathrm{kg}$$ hangs vertically. What is the correct expression for the acceleration $$a of $$the system?

Accepted Answer

$$a = \dfrac{m_2$ g - m_1$ g \sin 	heta}{m_1$ + m_2$}$$

Question 2

Given $$m_1 = 2.00$$\ \mathrm{kg}$$, m_2$ = 6.00\ \mathrm{kg}$$, and $$	heta = 55.0^\circ$$, what is the numerical value of the acceleration $$a of $$the system? (Take $$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$a = 5.8\ \mathrm{m/s^2$}$$

Question 3

For the same system, what is the tension $$T in $$the string connecting $$m_1$$ and $$m_2$$?

Accepted Answer

$$T = m_1$ (g \sin 	heta + a)$$

Question 4

If both masses start from rest, what is the speed $$v of $$each mass after they have moved $$2.00\ \mathrm{m}$$?

Accepted Answer

$$v = \sqrt{2 a d}$$

Question 5

A mass $$m_1 on a $$frictionless horizontal table is connected to a hanging mass $$m_2 by a $$light string over a pulley. Which equation correctly relates the acceleration $$a of $$the system to $$m_1$$ and $$m_2$$?

Accepted Answer

$$a = \dfrac{m_2$ g}{m_1$ + m_2$}$$

Question 6

In the horizontal table and hanging mass system, what is the tension $$T in $$the string?

Accepted Answer

$$T = m_1$ \dfrac{m_2$ g}{m_1$ + m_2$}$$

Question 7

If $$m_1 = 3.0$$\ \mathrm{kg}$$ and m_2$ = 2.0\ \mathrm{kg} in $$the horizontal table and hanging mass system, what is the acceleration $$a$$? ($$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$a = 2.8\ \mathrm{m/s^2$}$$

Question 8

A student claims that in the inclined plane system, the tension in the string is equal to the weight of $$m_1. Is $$this correct? Why or why not?

Accepted Answer

No, because the tension must also overcome the component of gravity along the incline and provide acceleration.

Question 9

If friction is introduced on the incline with coefficient $$\mu$$, how does the acceleration $$a$$ change?

Accepted Answer

$$a$$ decreases because friction opposes the motion.

Question 10

In the horizontal table and hanging mass system, if $$m_2 is $$much larger than $$m_1$$, what does the acceleration $$a$$ approach?

Accepted Answer

$$a$$ approaches $$g$$

Question 11

Which of the following best describes the role of the pulley in these systems?

Accepted Answer

It changes the direction of the tension force without adding friction.

Question 12

If the string in either system is not massless, how does this affect the acceleration calculation?

Accepted Answer

The acceleration decreases because the total mass increases.

Question 13

Suppose the incline is made steeper in the first system. What happens to the acceleration $$a$$?

Accepted Answer

$$a$$ increases because $$m_1 g$$ \sin 	heta$$ increases, reducing the net force opposing m_2$.

Question 14

If $$m_1$$ and $$m_2$$ are equal in the horizontal table and hanging mass system, what is the acceleration $$a$$?

Accepted Answer

$$a = \dfrac{g}{2}$$

Question 15

In the inclined plane system, if $$m_2 is $$not heavy enough to move $$m_1 up $$the incline, what can you conclude?

Accepted Answer

The net force is zero or negative, so the system does not accelerate.

Newton's Laws of Motion: Pulley Systems and Applications

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