# Direction of a rope with a ball inside an accelerating box

• UnPetitGarcon
In summary: So, in summary, the problem involves a crate with a mass of 180 kg sliding down a ramp inclined at an angle of 37° above the horizontal. Inside the crate, there is a person with a mass of 55 kg, and a steel washer is suspended from a light string attached to the top of the crate. The goal is to determine the coefficient of kinetic friction between the ramp and the crate. To solve this, a free body diagram is drawn for the crate and another one for the washer. From these diagrams, an equation is derived and it is determined that the correct diagram is the one with the tension in the same direction as the crate's acceleration. The presence of friction makes the washer's acceleration less compared to the situation
UnPetitGarcon
Homework Statement
Calculating the acceleration of a ball attached to a rope attached to the ceiling of a box accelerating along a ramp.
Relevant Equations
F=ma
A steel washer is suspended inside an empty shipping crate from a light string attached to the top of the crate. The crate slides down a long ramp that is inclined at an angle of 37° above the horizontal. The crate has mass 180 kg. You are sitting inside the crate(with a flashlight); your mass is 55 kg. As the crate is sliding down the ramp, you find the washer is at rest with respect to the crate when the string makes an angle of 68° with the top of the crate. What is the coefficient of kinetic friction between the ramp and the crate?

To solve this problem, I drew the free body diagram of the crate first. And from it I came up with an equation: sin(37°) - µcos(37°) = a/g, where a is the acceleration along the ramp and µ the coefficient of friction.

To solve for the a, I then drew the free body diagram of the ball. Here comes the problem: there are two possible directions for the tension.

I think both are possible. The second diagram is perfectly reasonable because the tension direction gives a positive value of force along the x direction. When this is combined with the weight in x direction, the net force causes the ball to accelerate down the ramp, which is what exactly happens in the problem. The first diagram is also ok because as long as the x component of the tension is smaller than the x component of the weight, the ball still accelerates down the ramp. So my question is: how to determine which one is correct.

In which direction does your crate slide here? Down the x-direction? In that case: In which direction does the washer have to accelerate?

UnPetitGarcon
mfb said:
In which direction does your crate slide here? Down the x-direction? In that case: In which direction does the washer have to accelerate?
Thanks for reply. Both the crate and the washer slide down the x-direction. But how does it related to the question?

UnPetitGarcon said:
Thanks for reply. Both the crate and the washer slide down the x-direction. But how does it related to the question?
What would be the angle without friction? Does the friction make the washer's acceleration more or less?

UnPetitGarcon
haruspex said:
What would be the angle without friction? Does the friction make the washer's acceleration more or less?
Got it! So Diagram 1 is the correct one because it makes the washer's acceleration less compared to the situation without friction. Thank you very much.

berkeman
UnPetitGarcon said:
Got it! So Diagram 1 is the correct one because it makes the washer's acceleration less compared to the situation without friction. Thank you very much.
Right.

## 1. How does the direction of a rope with a ball inside an accelerating box affect the motion of the ball?

The direction of the rope with a ball inside an accelerating box can greatly affect the motion of the ball. If the rope is pulled in the same direction as the acceleration, the ball will experience a greater force and accelerate in that direction. However, if the rope is pulled in the opposite direction of the acceleration, the ball will experience a smaller force and may even slow down or change direction.

## 2. What is the relationship between the tension in the rope and the acceleration of the box?

The tension in the rope is directly proportional to the acceleration of the box. This means that as the acceleration of the box increases, the tension in the rope also increases. Similarly, if the acceleration decreases, the tension in the rope will also decrease.

## 3. Can the ball inside an accelerating box ever experience zero tension in the rope?

Yes, it is possible for the ball inside an accelerating box to experience zero tension in the rope. This can occur when the acceleration of the box is equal to the acceleration due to gravity and the rope is pulled at an angle that cancels out the force of gravity on the ball.

## 4. How does the mass of the ball affect the tension in the rope?

The mass of the ball does not directly affect the tension in the rope. However, it does affect the acceleration of the ball, which in turn affects the tension in the rope. A heavier ball will require a greater force to accelerate, resulting in a higher tension in the rope.

## 5. How does the direction of the acceleration affect the direction of the tension in the rope?

The direction of the acceleration and the direction of the tension in the rope are always opposite to each other. For example, if the box is accelerating to the right, the tension in the rope will be pulling to the left. This is due to Newton's third law of motion, which states that every action has an equal and opposite reaction.

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