Tractor and pulleys - expression of speed

In summary: Va is the distance of the tractor (x), and l is the length of the rope. In summary, the problem involves a tractor lifting a bale with a system of pulleys. The goal is to find the expression for the bale's vertical speed in terms of the distance from the tractor to the pulley. The problem can be simplified by considering only kinematics and not forces. By breaking down the rope into three sections and considering the angles between them, it can be determined that the bale's vertical speed is half of the tractor's speed multiplied by the ratio of the distance from the tractor to the length of the rope.
  • #1
sofija
1
0

Homework Statement



A tractor named A is used to lift a bale B with a system of pulleys.
The tractor has an unknown speed called VA : find the expression of the bale's vertical speed VB in function of x (distance from the tractor to the pulley).
The diameter of both pulleys can be neglected.

Homework Equations



I call T the tension in the rope, N the reaction of the ground and W=mg the weight, mA the tractor's mass and mB the bale's mass, aA the tractor's acceleration and aB the bale's acceleration.

In A, I have : horizontally -T*(x/l) = mA * aA
vertically :N - mA*g + N*(h/l) = 0
In B : T - mB*g = mB*aB

The Attempt at a Solution



I've tried to put the problem into some equations, but I don't see how I can switch from accelerations to speed...moreover, I don't have any numeric data, so any piece of information is welcome!

Thanks in advance!
 

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  • #2
This is more of a trig problem than a physics problem. No need to consider forces. Hint: The rope doesn't stretch. Hint 2: How does VB depend on the rate that "l" increases?
 
  • #3
Hint: there's no dynamics involved in this problem: it is entirely kinematical. This means that we don't care about forces and so on: it is just "geometry": for a given position of the tractor, and a given length of the rope, the bale will be at a certain height.

Second hint: call the total rope length "L" and consider the 3 pieces of rope (going from the pulley to the bale, going from the bale up to the pulley again, and finally going from the pulley to the tractor) that make up the entire rope (and hence, whose sum must be equal to L).

EDIT: I see that Doc Al was typing the same thing while I was doing so...
 
  • #4
If I remember correctly, having this type of setup gives a force advantage to the tractor. Pulling rope (l) a distance of 10m will raise bail B by 5m. The force on (l) is equal to the force on each of ropes in the loop that make up (h-y), so an applied force of 10N on (l) will give a force of 20N on the bale.

First step is thinking what the equation would be with no angles:
Vb = (1/2)Va; where Vb and Va are parallel (the tractor would be going straight up at x=0)
Now realistically, maximum velocity of Vb is when Va is parallel to rope (l), so we'll use cosine (because cosine starts at maximum). Since we're using cosine we also need to use an angle that starts at 0 (where cosine is max). Sticking with this idea of max speed, Va will be parallel to rope (l), which will also be parallel to ground (x). Having a parallel between measurable distances is what we want for our angle of 0, so our cosine is for the angle between rope (l) and ground (x).
Vb = (1/2)(Va)cos(theta), alternatively written as:
Vb = (1/2)(Va)(x/l)
 
Last edited:

1. What is a tractor?

A tractor is a heavy-duty vehicle designed to provide high torque at low speeds, typically used for agricultural or construction purposes.

2. How do pulleys affect the speed of a tractor?

Pulleys are mechanical devices that use a wheel and rope or belt to transfer power and change the direction of force. In a tractor, pulleys can be used to increase or decrease the speed of the engine's rotation, which in turn affects the speed of the tractor's movement.

3. How are pulleys and speed related in a tractor?

The number and size of pulleys used in a tractor's transmission system directly impact the speed of the tractor. More pulleys or larger pulleys can increase the tractor's speed, while fewer or smaller pulleys can decrease it.

4. What is the expression of speed in relation to tractor and pulleys?

The expression of speed in relation to tractor and pulleys is known as gear ratio. This ratio compares the speed of the engine to the speed of the tractor's wheels. A higher gear ratio means the engine is rotating faster than the wheels, resulting in faster movement of the tractor.

5. Can the speed of a tractor be controlled by adjusting pulleys?

Yes, the speed of a tractor can be controlled by adjusting the size or number of pulleys in its transmission system. This allows for versatility in different working conditions and tasks, as the tractor's speed can be optimized for different levels of power and torque.

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