• Support PF! Buy your school textbooks, materials and every day products Here!

Torque of a rotating wheel on a static wheel

  • #1

Homework Statement


A uniform cylindrical wheel of mass ##m_{1}## and radius ##R_{1}## rotates with angular velocity ##\omega_{1}##. It lies a certain distance (along the same axis) from a static wheel of radius ##R_{2}## and mass ##m_{2}##. The wheels are then pushed against each other with a constant force ##F## uniformly distributed across the wheel's face. There is friction ##\mu## between the wheels.
qrnsz6.png

When the wheels first touch what is the magnitude of the torque that wheel 1 exerts on wheel 2 via friction?

Homework Equations


Split wheel 1 into infinitesimal concentric rings of radius ##r## and width ##dr## and find the torque exerted by one of these then integrate to find the total torque.

The Attempt at a Solution



I probably should use ##d\tau=dI \alpha## where ##dI## is the moment of inertia due to the cylindrical ring. But since ##\omega_{1}=const## it would seem that ##\alpha=\dot{\omega}_{1}=0##?
 

Attachments

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618

Homework Statement


A uniform cylindrical wheel of mass ##m_{1}## and radius ##R_{1}## rotates with angular velocity ##\omega_{1}##. It lies a certain distance (along the same axis) from a static wheel of radius ##R_{2}## and mass ##m_{2}##. The wheels are then pushed against each other with a constant force ##F## uniformly distributed across the wheel's face. There is friction ##\mu## between the wheels.
View attachment 232622
When the wheels first touch what is the magnitude of the torque that wheel 1 exerts on wheel 2 via friction?

Homework Equations


Split wheel 1 into infinitesimal concentric rings of radius ##r## and width ##dr## and find the torque exerted by one of these then integrate to find the total torque.

The Attempt at a Solution



I probably should use ##d\tau=dI \alpha## where ##dI## is the moment of inertia due to the cylindrical ring. But since ##\omega_{1}=const## it would seem that ##\alpha=\dot{\omega}_{1}=0##?
You are interested in torque exerted when they touch. The normal force pushing them together will cause a tangential force through friction that will exert the torque. Thing about using ##\tau = r \times F##.
 
  • #3
You are interested in torque exerted when they touch. The normal force pushing them together will cause a tangential force through friction that will exert the torque. Thing about using ##\tau = r \times F##.
Using ##d\tau=r\frac{F}{\pi R_{1}^{2}}\pi rdr## and then ##\tau=\int_{0}^{R_{1}}d\tau##?
 
  • #4
kuruman
Science Advisor
Homework Helper
Insights Author
Gold Member
8,705
2,117
Using ##d\tau=r\frac{F}{\pi R_{1}^{2}}\pi rdr## and then ##\tau=\int_{0}^{R_{1}}d\tau##?
Close. The surface of a ring of thickness ##dr## and radius ##r## is ##dA=2\pi r dr##.
 
  • #5
Dick
Science Advisor
Homework Helper
26,258
618
Close. The surface of a ring of thickness ##dr## and radius ##r## is ##dA=2\pi r dr##.
And ##F## is the normal force. The force acting against the rotation is the frictional force. There is a ##\mu## involved!
 
  • #6
Close. The surface of a ring of thickness ##dr## and radius ##r## is ##dA=2\pi r dr##.
Then ##d\tau=r\frac{F}{\pi R_{1}^{2}}2\pi rdr=2\frac{F}{R^{2}_{1}}r^{2}dr## and then ##\tau=\int_{0}^{R_{1}}d\tau=\frac{2}{3}R_{1}F##? The torque will be tangential to the surface and counterclockwise?
 
  • #7
And ##F## is the normal force. The force acting against the rotation is the frictional force. There is a ##\mu## involved!
The friction force torque will then be ##\tau=-\frac{2\mu}{3}R_{1}F_{1}## so that the total torque will be ##\tau=\frac{2}{3}R_{1}F_{1}(1-\mu)## in the same direction as ##\omega_{1}##?
 
  • #8
kuruman
Science Advisor
Homework Helper
Insights Author
Gold Member
8,705
2,117
The friction force torque will then be ##\tau=-\frac{2\mu}{3}R_{1}F_{1}## so that the total torque will be ##\tau=\frac{2}{3}R_{1}F_{1}(1-\mu)## in the same direction as ##\omega_{1}##?
Yes. Since the angular speed of the wheel is increasing, the torque on it and its angular velocity must be in the same direction.

Where does ##(1-\mu)## come from? What happens when ##\mu = 1##? Check your integral.
 
  • #9
Yes. Since the angular speed of the wheel is increasing, the torque on it and its angular velocity must be in the same direction.

Where does ##(1-\mu)## come from? What happens when ##\mu = 1##? Check your integral.
Not following. The ##(1-\mu)## comes from the torque of ##F## and the opposite torque of the friction force which is ##\mu F##. Am I missing ##2\pi## in the integral?
 
  • #10
kuruman
Science Advisor
Homework Helper
Insights Author
Gold Member
8,705
2,117
Newton's 3rd law says that to every action there is an equal and opposite reaction. In other words, you change the sign but not the magnitude. This applies to forces and, since torques are generated by forces, it applies to torques as well.
 
  • #11
Dick
Science Advisor
Homework Helper
26,258
618
Not following. The ##(1-\mu)## comes from the torque of ##F## and the opposite torque of the friction force which is ##\mu F##. Am I missing ##2\pi## in the integral?
##F## doesn't produce any net torque, contributions from opposite sides of the wheel cancel. Only the frictional force produces a net torque.
 
  • #12
##F## doesn't produce any net torque, contributions from opposite sides of the wheel cancel. Only the frictional force produces a net torque.
The torque will then be ##\frac{4\pi}{3}R_{1}\mu F##?
 
  • #13
Dick
Science Advisor
Homework Helper
26,258
618
The torque will then be ##\frac{4\pi}{3}R_{1}\mu F##?
Where did the '4' come from?
 
  • #14
Where did the '4' come from?
Kuruman mentioned my integral was missing something I thought I was missing a ##2\pi##. Perhaps he meant the factor ##1-\mu## should just be ##\mu##?
 
  • #15
Dick
Science Advisor
Homework Helper
26,258
618
Kuruman mentioned my integral was missing something I thought I was missing a ##2\pi##. Perhaps he meant the factor ##1-\mu## should just be ##\mu##?
You are throwing factors in because you are guessing. Go over it from the beginning and show what YOU think the result should be.
 
  • #16
You are throwing factors in because you are guessing. Go over it from the beginning and show what YOU think the result should be.
I think it should be ##\frac{2}{3}R_{1}\mu F##
 
  • #17
Dick
Science Advisor
Homework Helper
26,258
618
I think it should be ##\frac{2}{3}R_{1}\mu F##
And I think that's correct.
 
  • #18
kuruman
Science Advisor
Homework Helper
Insights Author
Gold Member
8,705
2,117
Kuruman mentioned my integral was missing something I thought I was missing a ##2\pi##. Perhaps he meant the factor ##1-\mu## should just be ##\mu##?
Yes, not knowing where the ##(1-\mu)## came from, I questioned whether it came as a result of faulty integration.
 

Related Threads for: Torque of a rotating wheel on a static wheel

Replies
4
Views
7K
Replies
1
Views
2K
  • Last Post
Replies
1
Views
3K
Replies
10
Views
628
Replies
25
Views
954
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
0
Views
1K
  • Last Post
Replies
1
Views
2K
Top