Centripetal acceleration problem

[bballcool34]

A small bug is sitting on the edge of a uniform disk of mass 2.5 kg and radius 20 cm initially at rest. There is a massless string wrapped around the disk and the coefficient of friction between the bug and the disk is u= 0.4. The string is then pulled with a constant tension of 4 N.

a) What is ther maximum linear speed the bug can have and stay on the disk?
b) How many seconds until the bug falls of the disk assuming the string is long enough to keep applying the force?

I know the solution, but I'm confused as to why my teacher did it. He said:

"Because there is a constant tangential acceleration and an increasing centripetal acceleration, so the acceleration is the resultant of those two perpendicular components.

For part a, He set potential energy equal to net force and solved for v (umg= ma). Why didn't he take into account centripetal force? Why is centripetal accel. increasing, but tangential acceleration constant?

For part b, he said v= a tangential * t. Why did he just use tangential acceleration to determine how long the bug stays on the disk?

Thanks in advance for any help.

 

[ehild]

The bug moves together with the disk along a circle. It sits at the rim of the disk, so its acceleration has a centripetal component v²/R. The rotation of the disk is accelerated by the tension of the string. You get the angular acceleration β by dividing the torque of the tension τ=TR with the moment of inertia of the disk (I): β =TR/I, and the linear acceleration at the rim is a_t=βR.

As the bug moves together with the disk, its acceleration also has centripetal and tangential components. The acceleration is the resultant of these two perpendicular accelerations, and the force is parallel to the resultant acceleration.

The magnitude of the resultant acceleration is

a=√(a_cp²+a_t²)

and the magnitude of the resultant force is ma.

The force on the bug is provided by the static friction, which maximum value is μgm: μgm=ma. Your teacher took the centripetal acceleration into account as 'a' is the resultant of both accelerations, centripetal and tangential.

As for b: the velocity is always tangential, and the bug stays at place while its speed is less than the maximum value. The rim moves with uniform tangential acceleration a_t, so the speed is proportional to the elapsed time: v=a_t*t .

ehild

 

[bballcool34]

Thanks for the reply.

One thing that is still confusing to me though is how, from the problem, I can know that tangential acceleration is constant, but centripetal acceleration is increasing?

 

[Doc Al]

One thing that is still confusing to me though is how, from the problem, I can know that tangential acceleration is constant, but centripetal acceleration is increasing?

Centripetal acceleration depends on the speed of the bug; tangential acceleration does not.

 

[rwisz]

"

I would like to clarify a few points about the solution provided by your teacher.

Firstly, the solution provided is correct in terms of finding the maximum linear speed and the time until the bug falls off the disk. However, there are a few important concepts that may have been overlooked or not fully explained.

Regarding the first question, it is important to understand that the bug is not moving in a straight line, but rather in a circular motion. This means that there are two types of acceleration acting on the bug - tangential acceleration (which is constant) and centripetal acceleration (which is increasing as the bug moves faster). The net acceleration is the resultant of these two components and it is what keeps the bug moving in a circular path.

Now, in order for the bug to stay on the disk, the net force acting on it must be directed towards the center of the disk. This is where the concept of centripetal force comes in. The centripetal force is the force that acts towards the center and keeps the bug moving in a circular path. In this case, the centripetal force is provided by the tension in the string. So, in order for the bug to stay on the disk, the tension in the string must be equal to the centripetal force.

Coming to the second question, the tangential acceleration is used to determine the time until the bug falls off the disk because it is the only component of acceleration that changes with time. The centripetal acceleration remains constant throughout the motion. Therefore, the tangential acceleration is used to determine the time taken for the bug to reach its maximum speed and fall off the disk.

In conclusion, while the solution provided by your teacher is correct, it is important to understand the underlying concepts of circular motion, centripetal force, and tangential acceleration to fully grasp the problem. I would suggest discussing these concepts with your teacher or seeking additional resources to clarify any confusion.