Question

A string is wrapped around a uniform disk of mass M = 1.2 kg and radius R = 0.07 m. (Recall that the moment of inertia of a uniform disk is (1/2)MR2.) Attached to the disk are four low-mass rods of radius b = 0.11 m, each with a small mass m = 0.5 kg at the end. The device is initially at rest on a nearly frictionless surface. Then you pull the string with a constant force F = 29 N. At the instant when the center of the disk has moved a distance d = 0.039 m, a length w = 0.023 m of string has unwound off the disk.(a) At this instant, what is the speed of the center of the apparatus?v = ? m/s(b) At this instant, what is the angular speed of the apparatus?w1 = ? radians/s(c) You keep pulling with constant force 29 N for an additional 0.022 s. Now what is the angular speed of the apparatus?w2 = ? radians/s

Answer 1

Explanation:

Given that,

Mass of disk = 1.2 kg

Radius = 0.07 m

Radius of rod = 0.11 m

Mass of small disk = 0.5 kg

Force = 29 N

Time t = 0.022 s

$\theta=0.023\ m$

Distance d= 0.039 m

(I). We need to calculate the speed of the apparatus

Using work energy theorem

$W=\Delta K.E$

$Fd=\dfrac{1}{2}mv^2$

$v=\sqrt{\dfrac{2Fd}{M+4m}}$

Where, m = total mass

v = velocity

F = force

d = distance

Put the value into the formula

$v=\sqrt{\dfrac{2\times29\times0.039}{1.2+4\times0.5}}$

$v=0.840\ m/s$

(b). We need to calculate the angular speed of the apparatus

Using formula of torque

$\tau=I\alpha$

$F\times=(\dfrac{1}{2}MR^2+4mb^2)\alpha$

$29\times0.07=(\dfrac{1}{2}\times1.2\times0.07^2+4\times0.5\times0.11^2)\alpha$

$\alpha=\dfrac{29\times0.07}{0.02714}$

$\alpha=74.79\ rad/s^2$

We need to calculate the angular speed of the apparatus

Using equation of angular motion

$\omega=\omega_{0}+\alpha t$

Put the value into the formula

$\omega=0+74.79\times0.022$

$\omega=1.645\ rad/s$

(c).  We need to calculate the angular speed of the apparatus

Using equation of angular motion

$\omega_{0}^2=\omega^2+2\alpha t$

Put the value into the formula

$\omega_{0}^2=1.645^2+2\times74.79\times0.022$

$\omega=2.44\ rad/s$

Hence, This is required equation.