Q: For the scenario where the tractor moves at a constant speed of 4 m/s and $$s_A = 5\ \mathrm{m}$$, what is the tension in the rope? (Assume $$g = 9.81\ \mathrm{m/s^2$}$$ and neglect friction.)

$$T = 150 \times 9.81\ \mathrm{N}$$

Question 1

A 150-kg load B is lifted by a 24-m-long rope, boom, and pulley system as shown. If the tractor moves to the right at a constant speed of 4 m/s and $$s_A = 5\ \mathrm{m}$$, what is the correct expression for the length of rope between the top of the boom and the tractor at this instant?

Accepted Answer

$$L = \sqrt{(12\ \mathrm{m})^2$ + (s_A)^2$}$$

Question 2

Given the system in the image, if the tractor moves to the right at a constant speed, what is the vertical velocity of the load B when $$s_A = 5\ \mathrm{m}$$?

Accepted Answer

$$v_B = \frac{s_A}{\sqrt{(s_A)^2$ + (12)^2$}} v_A$$

Question 3

For the scenario where the tractor moves at a constant speed of 4 m/s and $$s_A = 5\ \mathrm{m}$$, what is the tension in the rope? (Assume $$g = 9.81\ \mathrm{m/s^2$}$$ and neglect friction.)

Accepted Answer

$$T = 150 	imes 9.81\ \mathrm{N}$$

Question 4

If the tractor accelerates to the right at $$3\ \mathrm{m/s^2$}$$ and has a velocity of $$4\ \mathrm{m/s} at$$ $$s_A = 5\ \mathrm{m}$$, what additional force must the rope provide to the load B, aside from its weight?

Accepted Answer

The rope must provide an upward force equal to $$m a_B$$, where $$a_B is $$the vertical acceleration of B.

Question 5

Which equation correctly relates the change in rope length to the positions $$s_A$$ and $$s_B as $$the tractor moves?

Accepted Answer

$$\sqrt{(s_A)^2$ + (12)^2$} + s_B = 24$$

Question 6

If $$s_A = 5\ \mathrm{m}$$, what is the value of $$s_B$$ given the total rope length is 24 m and the vertical height is 12 m?

Accepted Answer

$$s_B = 24 - \sqrt{(5)^2$ + (12)^2$}$$

Question 7

When the tractor is accelerating, which of the following best describes how to find the vertical acceleration of the load B?

Accepted Answer

Differentiate the constraint equation twice with respect to time.

Question 8

If the tractor moves to the right at a constant speed, what is the acceleration of the load B?

Accepted Answer

Zero

Question 9

If the tension in the rope is measured to be greater than $$150 	imes 9.81\ \mathrm{N}$$, what can be inferred about the motion of the load B?

Accepted Answer

The load is accelerating upward.

Question 10

Suppose the tractor suddenly stops while the load is in motion. What happens to the tension in the rope immediately after stopping?

Accepted Answer

The tension drops to just support the weight of the load.

Question 11

If the rope is not massless, how would this affect the calculation of tension in the rope?

Accepted Answer

The tension would vary along the length of the rope, being greatest at the top.

Question 12

If the tractor moves to the right at a constant speed, what is the power delivered by the rope to the load B?

Accepted Answer

$$P = T v_B$$

Question 13

Which of the following best describes the role of the pulley in this system?

Accepted Answer

It changes the direction of the force applied by the rope.

Question 14

If the tractor moves to the right with a velocity $$v_A$$ and the load rises with velocity $$v_B$$, which equation relates $$v_A$$ and $$v_B$$?

Accepted Answer

$$v_B = \frac{s_A}{\sqrt{(s_A)^2$ + (12)^2$}} v_A$$

Question 15

If the load B is 150 kg and the acceleration due to gravity is $$9.81\ \mathrm{m/s^2$}$$, what is the minimum tension required in the rope to keep the load stationary?

Accepted Answer

$$T = 150 	imes 9.81\ \mathrm{N}$$

Dynamics of a Pulley System: Tension in a Rope with Moving Tractor

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