Aditya's challenges in Mechanics 8
The arrangement shown above consists of a solid cylinder and a solid sphere, each of mass , on which a light thread is wound. Find the tension in the thread in the process of motion.
The answer is of the form for coprime positive integers and . Find .
Details and Assumptions:
-
The radii of of the cylinder and sphere are equal and are equal to
-
Take acceleration due to gravity as .
-
The friction in the axle of the upper cylinder is assumed to be absent.
-
There is no slipping anywhere.
The answer is 31.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
Let T be the tension in the string.
On the cylinder, the torque is provided by the tension.
Hence, the torque equation is: T r = 2 m r 2 α 1
Therefore, α 1 = m r 2 T
Now, on the sphere, there is a torque provided by the tension.
Hence, the torque equation is: T r = 5 2 m r 2 α 2
Therefore, α 2 = 2 m r 5 T
Now, we balance the forces on the sphere.
m g − T = m a
Since there is pure rolling, a = ( α 1 + α 2 ) r
Therefore, on substituting and inserting the values, we get: T = 1 1 2 0