All the answers are given, I just need step by step solutions please. 1. A 50 kg
ID: 2243117 • Letter: A
Question
All the answers are given, I just need step by step solutions please.
1. A 50 kg cylinder of radius 20 cm starts from rest and reaches 2.0 revolutions per second after 2.0 seconds. Assume the angular acceleration is constant during this time.
a) What is the angular acceleration? (2pi rad/s^2)
b) Through what angle did it turn? (4pi rad)
c) What is the moment of inertia about a vertical axis passing through its centre? (1 kgm^2)
d) What torque must have been applied to the cylinder during the 2.0 seconds? (2pi Nm)
e) At the end of the 2.0 seconds, what is the rotational kinetic energy? (8pi^2 J)
2. As the earth goes round the sun, its orbit is approximately circular.
a) What is its angular velocity? (2.0 x 10^-7 rad/s)
b) What is its angular acceleration? (0)
c) What is its acceleration? (5.95 x 10^-3 m/s^2)
d) What is its speed? (30 km/s)
Explanation / Answer
1) Given 2 revolutions per second after 2 seconds from rest .
a) one revolution = 2 pi radians
initial angular velocity wi = 0
final angular velocity wf=2 rev/s= 4 pi rad / s
time = 2 sec
according to the equation
wf= wi + alpha t
where alpha is the angular acceleration .
4pi =0 + 2 alpha
alpha = 2 pi rad /s2
SO angular acceleration is 2pi rad/s2
b) initial angle =0
aacording to the equation
theta final - theta initial = wi t + 1/2 * alpha t2
theta f -0 =0 + 1/2 * 2pi * 4
theta f = 4 pi
So the angle turned is 4pi radians
c) Moment of inertia of a cylinder about a vertical axis passing through its centre is given by
I = 1/2 M R2
Given M =50 kg
R= 0.2 m
Moment of inertia I = 1/2 * 50 * (0.2)2
I= 1 kg/m2
d) Torque = moment of inertia x angular acceleration
T= I * alpha
T = 1 x 2pi
T=2pi
Torque is 2pi Nm
e) K.E = 1/2 * I * (wf)2
K.E = 1/2 * 1 * ( 4pi)2
K.E = 8 pi2 J
2) Time Period of earth
T= 365.25 days
T=365.25 x 24 x 60 x 60 sec
we know angular velocity w = 2pi / T
w= 2pi / 365.25 x 24 x 60 x 60
w= 1.99 x 10-7 rad /s
angular velocity is 1.99 x 10-7 rad/s
b) angular acceleration is the rate of change of angular velocity .As earth rotates around the sun with a constant angular velocity ,there is no change in it .
SO the angular acceleration is zero .
d) we know that Velocity V = rw
r is the distance between sun and earth
r= 1496 x 105 km
V= 1496 x 105 x 1.99 x 10-7
V = 29.92 km/s
c) acceleration a = v2/ r
a= (29.92 x 103 )2 / 1496 x 108
a= 5.984 x 10-3 m/s2
acceleration is 5.984 x 10-3 m/s2