Name: Inertial Frames of Reference
Uploaded: 2019-07-19
Duration: 7 min 10 s
Description: Imagine if you are on a bus or a train, can you say you are in an inertial frame of reference? If yes, how can you predict it? Let us explore more in this vi...

Question

A uniform cylinder of radius $R$ is spinned about its axis to the angular velocity $ω_{0}$ and then placed in a corner. The coefficient of friction between the corner walls and the cylinder is equal to $k$ . How many turns will the cylinder accomplish before it stops?

Accepted Answer

The net force on the rigid body is zero, as a result, it is in translational equilibrium. The frictions acting on the rigid body are kinetic, which retards the rigid body and finally stops it after some angular rotation. Using the free body diagram of the rigid body and the concept of equilibrium for the cylinder,

Force equilibrium in vertical direction:

$m g = N_{1} + k N_{2} . . . (1)$

Force equilibrium in horizontal direction:

$N_{2} = k N_{1} . . . (2)$

Solving equations $(1)$ and $(2)$ , we get,

$N_{1} = \frac{m g}{1 + k^{2}}$ and $N_{2} = k \frac{m g}{1 + k^{2}}$

If the net torque on the cylinder is $τ_{z}$ , angular acceleration is $α_{z}$ and the moment of inertia is $I$ , then

$τ_{z} = I α_{z}$

or, $- k N_{1} R - k N_{2} R = \frac{m R^{2}}{2} α_{z}$

or, $- \frac{k m g R (1 + k)}{1 + k^{2}} = \frac{m R^{2}}{2} α_{z}$ [putting the values of $N_{1} & N_{2}]$

or, $α_{z} = - \frac{2 k (1 + k) g}{(1 + k^{2}) R}$

Now, using third equation of kinematics of rotation, we get

$ω^{2} = ω_{0}^{2} + 2 α_{z} Δφ$

Here, final angular velocity, $ω = 0$ , initial angular velocity is $ω_{0}$ and angular displacement is $∆ φ$

$Δφ = \frac{ω_{0}^{2} (1 + k^{2}) R}{4 k (1 + k) g}$ [putting the value of $α_{z}$ and simplifying]

As cylinder rotates by angle $2 π$ in one revolution, so the required number of turns before it stops is given by

$n = \frac{Δφ}{2 π} = \frac{ω_{0}^{2} (1 + k^{2}) R}{8 π k (1 + k) g}$

A uniform cylinder of radius $R$ is spinned about its axis to the angular velocity $ω_{0}$ and then placed in a corner. The coefficient of friction between the corner walls and the cylinder is equal to $k$ . How many turns will the cylinder accomplish before it stops?

Important Questions on PHYSICAL FUNDAMENTALS OF MECHANICS

Learn and Prepare for any exam you want !

A uniform cylinder of radius R is spinned about its axis to the angular velocity ω0 and then placed in a corner. The coefficient of friction between the corner walls and the cylinder is equal to k. How many turns will the cylinder accomplish before it stops?

Important Questions on PHYSICAL FUNDAMENTALS OF MECHANICS

Learn and Prepare for any exam you want !

A uniform cylinder of radius $R$ is spinned about its axis to the angular velocity $ω_{0}$ and then placed in a corner. The coefficient of friction between the corner walls and the cylinder is equal to $k$ . How many turns will the cylinder accomplish before it stops?