Pulley Problem: Find acceleration of block

In summary: Also, the equation for tension should be T=ma+mg. In summary, the problem involves a 1.5 kg block and a 2.5 kg block attached to opposite ends of a light rope, which hangs over a solid, frictionless pulley with a diameter of 30 cm and a mass of 0.75 kg. Using Newton's second law and the equation for torque, the net torque can be found to be equal to T1r + T2r, where T1 and T2 are the tensions on the rope attached to the blocks and r is the radius of the pulley. The torque depends on the direction of rotation, determined by the direction of acceleration, which is found to be negative due
  • #1
sona1177
173
1

Homework Statement


A 1.5 kg block and a 2.5 kg block are attached to opposite ends of a light rope. The rope hangs over a solid, frictionless pulley that is 30 cm in diameter and has a mass of .75 kg. When the blocks are released, what is the acceleration of the lighter rock?


Homework Equations



alpha= net torque/moment of intertia
F=ma



The Attempt at a Solution



I know Newton's law for the 1.5 kg block is T-W=ma and for the 2.5 kg block is T-W=-ma now the problem is the pulley. How do I calculate this torque with the signs correctly? Won't this vary depending on how I draw the diagram? Since the pulley has mass, the tensions on the both sides of the rope are not equal. Can someone please help me? I don't know how to go about finding the torque for the pulley.
 
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  • #2
Why don't you make the rope as your axis? That's how I always approached pulley problems. Linear acceleration to the right is positive, while to the left is negative.

For torque, counterclockwise is positive alpha, clockwise is negative alpha.
 
  • #3
thrill3rnit3 said:
Why don't you make the rope as your axis? That's how I always approached pulley problems. Linear acceleration to the right is positive, while to the left is negative.

I am afraid I don't understand what you mean by "make the rope as your axis". I always thought a force that makes the pulley go counter clockwise is positive but clockwise is negative. Your help is greatly appreciated :)
 
  • #4
Your linear acceleration eq is wrong

1.5 kg block is T-W=ma and for the 2.5 kg block is T-W=-ma

so would you have ma=-ma?

Check your diagram. I was just suggesting another way of setting up the axes, but it's totally up to you.
 
  • #5
thrill3rnit3 said:
Your linear acceleration eq is wrong



so would you have ma=-ma?

Check your diagram. I was just suggesting another way of setting up the axes, but it's totally up to you.

Why is it wrong? The 2.5 kg block is accelerating down so therefore I thought its acceleration is negative.
 
  • #6
edit: nevermind everything I just said. I thought T stood for the same unknown.
 
  • #7
Anyways, just continue with what you were doing. Counter clockwise rotation produces positive torque, clockwise rotation produces negative torque.

and a=alpha(radius), so that should give you the link between alpha and a
 
  • #8
That is my question I don't know the signs of the torques. Are they both clockwise ?
 
  • #9
Make an assumption. Do you suppose that the rope will slide left or right? After you make the initial assumption, just make sure everything else remains consistent.

You won't make a mistake if everything that follows is consistent. If you get a negative answer, then that means it's actually sliding the other way.
 
  • #10
Ok I changed my mind so that the 2.5 kg block is on the left and the 1.5 kg block is on the right. This means the rope Is moving towards the right (counter clockwise). This means:

Net torque= T1r + T2r where capital T1 stands for the tension of the rope attached to the 2.5 kg block and T2 for the tension of the rope attached to the 1.5 kg block. And I=.5MR^2

So alpha = 2 (T1 + T2)/MR where capital M stands for the mass of the pulley. This means linear acceleration is alpha * r therefore it is = 2(T1 + T2)/M and by Newton's second law since T1= -m1a + w1 then a = 2(-m1a + w1 + T2)/M so T2=(Ma/2) + m1a -w1. Plugging this into T2-w2=m2a for T2 and doing algebra gives 28.5 m/s. This is wrong :( can someone please help?
 
  • #11
sona1177 said:
Net torque= T1r + T2r

The right rope is torquing it clockwise [ negative ], the left rope is torquing it counterclockwise [ positive ], and you assumed that the system is moving right -> clockwise -> negative alpha.

-T1r+T2r= - Ialpha
 
Last edited:

What is a pulley and how does it work?

A pulley is a simple machine that consists of a wheel with a groove around its circumference and a rope or cable that runs through the groove. It is used to lift or lower objects by changing the direction of the force applied. When a force is applied to one end of the rope, the other end moves in the opposite direction, causing the object to move up or down.

What is the acceleration of a block in a pulley system?

The acceleration of a block in a pulley system depends on several factors, including the mass of the block, the mass of the pulley, and the tension in the rope. In order to find the acceleration, you need to apply Newton's second law (F=ma) and consider all the forces acting on the block, such as the weight of the block, the tension in the rope, and any friction present.

How do you solve a pulley problem to find the acceleration of a block?

To solve a pulley problem and find the acceleration of a block, you will need to draw a free-body diagram of the system and identify all the forces acting on the block. Then, apply Newton's second law to set up an equation and solve for the acceleration. You may also need to use other principles, such as the conservation of energy or the equations of motion, depending on the specific problem.

What are some common mistakes when solving a pulley problem?

Some common mistakes when solving a pulley problem include forgetting to consider all the forces acting on the block, not correctly setting up the equations, and using the wrong units. It is important to carefully analyze the problem and double-check your calculations to avoid these errors.

What are some real-life applications of pulleys?

Pulleys are used in a variety of real-life applications, such as elevators, cranes, and flagpoles. They are also commonly used in household items, such as window blinds and garage doors. In addition, pulleys are used in many industrial and manufacturing processes, such as conveyor belts and assembly line machines.

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