Minimum linear velocity attained by sphere

[utkarshakash]

Homework Statement

A sphere of mass M and radius R is moving on a rough fixed surface, having co-efficient of friction μ, with a velocity v towards right and angular velocity ω clockwise. It will attain a minimum linear velocity at time (take v>ωR)

The Attempt at a Solution

Since v>ωR the sphere rolls with slipping. So frictional force will act in the backward direction. Using the equation [itex]\int \tau dt = \int dL [/itex] where τ=μmgR.

[itex]\mu mgRt= \frac{2}{5} mR^2 (\omega ' - \omega) \\

\mu mg = m \frac{dv}{dt} \\

\mu gt = (v' - v)
[/itex]

Using the relation v'=ω'R and solving the above two equations I get
t= 2(v-ωR)/3μg. But the correct answer has 7 in the denominator.

 

[mfb]

[itex]\mu mg = m \frac{dv}{dt}[/itex]

There is a minus sign missing.

I don't see why this point is called "minimum linear velocity" - it is the point where the sphere stops slipping.

 

[utkarshakash]

There is a minus sign missing.

I don't see why this point is called "minimum linear velocity" - it is the point where the sphere stops slipping.

Why there should be a minus sign? Since friction acts backwards, so is the acceleration. Hence both minus cancels out. Are you trying to say that since velocity is decreasing that's why dv/dt should carry a '-' with it?

 

[mfb]

Acceleration is backwards, but as you write your equation both sides are positive, so v increases. You can use a negative v everywhere, but then things get really confusing and the initial v>ωR does not work.

 

[Nickie]

I would first check the equations and calculations to ensure that they are accurate and relevant to the given scenario. I would also consider other factors that may affect the minimum linear velocity attained by the sphere, such as air resistance or the surface roughness. Furthermore, I would conduct experiments or simulations to verify the results and make any necessary adjustments. It is important to critically evaluate and analyze all aspects of the problem in order to provide a scientifically sound response.