Finding Acceleration Of A Bucket Moving Downwards With A Pulley

[Lost_Prophet75]

Homework Statement

A 26.5 kg block is connected to an empty 1.00 kg bucket by a cord running over a frictionless pulley (Fig. 4-57). The coefficient of static friction between the table and the block is 0.435 and the coefficient of kinetic friction between the table and the block is 0.320. Sand is gradually added to the bucket until the system just begins to move.

Homework Equations

So far I have solved the mass of the sand added to the bucket, which is 10.528kg, ignoring the mass of the bucket. I am still unsure of how to setup the equation, because I am confused on what I use for tension. The tension I originally used should have changed due to the mass of the sand now added to the bucket. The first time I solved for tension, I didn't need acceleration, as the bucket was at rest. Every equation I try to set up comes out needing two variables, neither of which I have any idea how to solve for.

The Attempt at a Solution

When I try to find tension, my equation comes out to one of these two (Sorry if I write these wrong, still figuring out how to type out an equation).

For the Tension in the block, i get T=$$\mu$$N + ma, which requires the use of either tension or acceleration to solve. I got this equation by using a free-body diagram, which had weight going down the y-axis, normal force going up the y-axis, tension to the right on the x-axis, and Friction going left on the x-axis. The sum of my forces would then be Tension minus friction equals mass times acceleration, so I moved friction over, and converted it to mu times normal force.

My Other equation is for the bucket, which is T=mg +ma. For this one, ny free body has two forces acting on it, both on the y-axis. Tension going up, weight moving it down. My sum of forces came out to be Tension minus weight equals mass times acceleration.

What am I missing here, or am I over complicating things?

 

[usfsurfer]

Ok, If I'm understanding what it is you're trying to solve for, it is the tension in the line between the two blocks. You have already correctly calculated the additional sand mass that must be added into the bucket to overcome the static friction, so we are now working with a kinetic friction acting on the block located on the table.

To get to the tension, you first want to solve for what the acceleration of the two bodies is together (this can be done without breaking it down into separate free body diagrams knowing that the tension in the line must be constant). You can then use that acceleration and go back to the free body diagram of either block to solve for the tension in the cord.

This is my first post on here and I'm not too sure of how much information is normally given (I don't want to just hand the solution over). I hope this can at least send you in the right direction. Let me know if you need more help.

 

[m2003]

First of all, it is great that you have already solved for the mass of the sand added to the bucket. That is an important step in solving this problem.

In order to find the acceleration of the bucket moving downwards with the pulley, you will need to use the equations of motion. These equations are:

1. v = u + at (where v is final velocity, u is initial velocity, a is acceleration, and t is time)
2. s = ut + 1/2at^2 (where s is displacement)
3. v^2 = u^2 + 2as (where v is final velocity, u is initial velocity, a is acceleration, and s is displacement)

In this case, the acceleration you are looking for is the acceleration of the bucket. Therefore, you will need to consider the forces acting on the bucket.

As you mentioned, there are two forces acting on the bucket - tension and weight. The weight of the bucket can be calculated as mg, where m is the mass of the bucket and g is the acceleration due to gravity (9.8 m/s^2).

The tension in the cord can be found by considering the forces acting on the block. As you correctly identified, the forces acting on the block are weight, normal force, tension, and friction. The normal force can be calculated as N = mg, where m is the mass of the block. The friction force can be calculated as F = μN, where μ is the coefficient of kinetic friction and N is the normal force.

Now, using these values, you can set up an equation of motion for the bucket by considering the forces acting on it. The equation will be:

T - mg = ma

where T is the tension in the cord, m is the mass of the bucket, g is the acceleration due to gravity, and a is the acceleration of the bucket.

You can then use this equation to solve for the acceleration of the bucket (a). Once you have found the acceleration, you can use it to solve for the final velocity (v) and the displacement (s) of the bucket using the equations of motion.

I hope this helps clarify the process for finding the acceleration of the bucket moving downwards with the pulley. Remember to always consider the forces acting on the object you are trying to find the acceleration for, and use the appropriate equations of motion to solve for it. Good luck!