Minimum Angular Speed to start SLIPPING BEFORE ROLLING WITHOUT SLIPPING

In summary, the problem asks for the minimum angular speed w to start slipping before rolling without slipping, as well as the time it takes for the cylinder to start rolling without slipping if w is adequate. However, there is no clear solution to this problem as it is unclear what the initial conditions are and how the cylinder is moving when it first contacts the surface.
  • #1
Ransalu
1
0

Homework Statement


Minimum Angular Speed to start SLIPPING BEFORE ROLLING WITHOUT SLIPPING.


Homework Equations


A cylinder of mass M with radius r is given an angular speed of an w about an axis, parallel to its length , which passes through its centre. The cylinder is gently lowered onto an upward direction of inclined frictional surface which makes an angle alpha with the horizontal axis. The coefficient of friction of the two surfaces is u.
*What is the minimum angular speed w to start slipping before rolling without slipping?
*If the angular speed w is adequate to start slipping how long does it take before the cylinder starts to roll without slipping?



The Attempt at a Solution


I can see no possible way to reach the first part of the question. I can't realize why does the cylinder slip firstly before purely rolling. Is it slipping with rolling or without rolling at the begining?
 
Physics news on Phys.org
  • #2
Usually you begin this kind of problem by finding the maximum up-the-hill force you can get - which is the friction force minus the component of weight down the hill. Then use F=ma to get the maximum acceleration. Alas, this problem doesn't ask for the max acceleration, but rather the max speed. If the cylinder is not moving tangentially when it contacts the surface, any w without slipping will require infinite acceleration at the moment of first contact with the surface as its speed changes from zero to rw instantly. It can't have that so it will slip.

If we interpret the "touches gently" as a cylinder that is moving in the direction uphill along the ramp just before it touches then no acceleration is required to continue moving up the ramp at that same w or rw. So it won't slip.

Note that the cylinder must use up its rotational energy to climb the hill. As it goes up, it will decelerate. I don't see how this gives us any grip on the problem, either.

Looks like an impossible problem to me, too. I will be most interested in seeing a solution!
 
  • #3


I would like to clarify the question before attempting to provide a response. It is unclear what is meant by "slipping before rolling without slipping." It is possible that the question is asking for the minimum angular speed at which the cylinder will start slipping without rolling, or it could be asking for the minimum angular speed at which the cylinder will start rolling without slipping. These are two different scenarios with different solutions.

Assuming the question is asking for the minimum angular speed at which the cylinder will start slipping without rolling, the answer can be found by considering the forces acting on the cylinder. The normal force, N, acting on the cylinder is given by N = Mgcosα, where M is the mass of the cylinder, g is the acceleration due to gravity, and α is the angle of the inclined surface. The frictional force, F, is given by F = μN, where μ is the coefficient of friction.

For the cylinder to start slipping without rolling, the frictional force must be equal to or greater than the maximum static frictional force, which is given by Fmax = μsN, where μs is the coefficient of static friction. Therefore, we can write the following equation:

F ≥ Fmax

μN ≥ μsN

μ ≥ μs

This means that the minimum coefficient of friction required for the cylinder to start slipping without rolling is equal to the coefficient of static friction.

Now, we can consider the torque acting on the cylinder. The torque, τ, is given by τ = Iα, where I is the moment of inertia of the cylinder and α is the angular acceleration. The moment of inertia for a cylinder rotating about its central axis is given by I = ½Mr^2.

For the cylinder to start slipping without rolling, the torque due to the frictional force must be greater than the torque due to the normal force. We can write the following equation:

τF ≥ τN

Fμr ≥ Nwr

μ ≥ wr

Substituting in the expression for the minimum coefficient of friction, we get:

μs ≥ wr

Therefore, the minimum angular speed at which the cylinder will start slipping without rolling is given by w = μs/r.

For the second part of the question, we can use the equation τF = Nwr to find the time it takes for the cylinder to start rolling without slipping. We can rearrange this equation
 

What is the concept of Minimum Angular Speed?

The Minimum Angular Speed refers to the minimum rotational speed required for an object to start slipping before rolling without slipping. It is dependent on the mass, shape, and surface properties of the object.

Why is knowing the Minimum Angular Speed important?

Knowing the Minimum Angular Speed is important in understanding the behavior of rotating objects, such as wheels or gears. It helps in predicting when an object will start slipping and how much force is needed to prevent slipping.

How is the Minimum Angular Speed calculated?

The Minimum Angular Speed can be calculated using the formula ω = v/r, where ω is the angular speed, v is the linear speed, and r is the radius of the object. It can also be calculated using the equation ω = √(μg/r), where μ is the coefficient of friction and g is the acceleration due to gravity.

What factors affect the Minimum Angular Speed?

The Minimum Angular Speed is affected by the mass, shape, and surface properties of the object. It is also influenced by external factors such as the coefficient of friction, the type of surface the object is rolling on, and the force applied to the object.

What happens if the Minimum Angular Speed is exceeded?

If the Minimum Angular Speed is exceeded, the object will start slipping and will not be able to roll without slipping. This can cause the object to lose its stability and potentially lead to accidents or damage to the object.

Similar threads

  • Introductory Physics Homework Help
Replies
4
Views
1K
  • Introductory Physics Homework Help
Replies
21
Views
1K
Replies
39
Views
1K
  • Introductory Physics Homework Help
3
Replies
97
Views
3K
  • Introductory Physics Homework Help
2
Replies
35
Views
2K
  • Introductory Physics Homework Help
Replies
7
Views
2K
  • Introductory Physics Homework Help
Replies
14
Views
1K
  • Introductory Physics Homework Help
Replies
8
Views
2K
  • Mechanics
Replies
7
Views
2K
  • Introductory Physics Homework Help
Replies
2
Views
2K
Back
Top