Question

A torque of 36.5 N · m is applied to an initially motionless wheel which rotates around a fixed axis. This torque is the result of a directed force combined with a friction force. As a result of the applied torque the angular speed of the wheel increases from 0 to 10.3 rad/s. After 6.10 s the directed force is removed, and the wheel comes to rest 60.6 s later.

(a) What is the wheel's moment of inertia (in kg m2)? kg m
(b) What is the magnitude of the torque caused by friction (in N m)? N m
(c) From the time the directed force is initially applied, how many revolutions does the wheel go through?
______ revolutions

Answer 1

`21.6\ \text{kg m}^2`

`3.672\ \text{Nm}`

`54.66\ \text{revolutions}`

Step-by-step explanation:

`\tau` = Torque = 36.5 Nm

`\omega_i` = Initial angular velocity = 0

`\omega_f` = Final angular velocity = 10.3 rad/s

t = Time = 6.1 s

I = Moment of inertia

From the kinematic equations of linear motion we have

`\omega_f=\omega_i+\alpha_1 t\\\Rightarrow \alpha_1=(\omega_f-\omega_i)/(t)\\\Rightarrow \alpha_1=(10.3-0)/(6.1)\\\Rightarrow \alpha_1=1.69\ \text{rad/s}^2`

Torque is given by

`\tau=I\alpha_1\\\Rightarrow I=(\tau)/(\alpha_1)\\\Rightarrow I=(36.5)/(1.69)\\\Rightarrow I=21.6\ \text{kg m}^2`

The wheel's moment of inertia is
`21.6\ \text{kg m}^2`

t = 60.6 s

`\omega_i` = 10.3 rad/s

`\omega_f` = 0

`\alpha_2=(0-10.3)/(60.6)\\\Rightarrow \alpha_1=-0.17\ \text{rad/s}^2`

Frictional torque is given by

`\tau_f=I\alpha_2\\\Rightarrow \tau_f=21.6* -0.17\\\Rightarrow \tau=-3.672\ \text{Nm}`

The magnitude of the torque caused by friction is
`3.672\ \text{Nm}`

Speeding up

`\theta_1=0* t+(1)/(2)* 1.69* 6.1^2\\\Rightarrow \theta_1=31.44\ \text{rad}`

Slowing down

`\theta_2=10.3* 60.6+(1)/(2)* (-0.17)* 60.6^2\\\Rightarrow \theta_2=312.03\ \text{rad}`

Total number of revolutions

`\theta=\theta_1+\theta_2\\\Rightarrow \theta=31.44+312.03=343.47\ \text{rad}`

`(343.47)/(2\pi)=54.66\ \text{revolutions}`

The total number of revolutions the wheel goes through is
`54.66\ \text{revolutions}`.