Solving Angular Momentum Problems with Linear & Angular Acceleration

AI Thread Summary
The discussion focuses on solving angular momentum problems involving a sphere that transitions from sliding to rolling. Key points include the relationship between linear speed (V_{cm}) and angular speed (ω), with the condition that sliding stops when V_{cm} equals ω times the radius (r). The kinetic friction force affects both linear and angular acceleration, causing the ball to decelerate while increasing its rotational speed. The equations governing these motions involve the moment of inertia and the frictional force, which is determined by the coefficient of kinetic friction (μ_k) and the weight of the ball. Understanding these dynamics is crucial for calculating the duration and distance of sliding, as well as the final linear speed when rolling begins.
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I know this sounds weird, but I studied QM and GR before classical physics, and I'm just lost when it comes to angular momentum problems.

Homework Statement


A sphere is moving along a lane. It slides initially then rolls. The initial speed is V_{cm} and initial angular speed \omega. The coefficient of kinetic friction between the ball and the lane is also known. The kinetic frictional force acting on the ball causes a linear acceleration of the ball while producing a torque that causes an angular acceleration of the ball. When V_{cm} has decreased enough and \omega has increased enough, the ball stops sliding and rolls smoothly.

Given: V_{cm} initial, \omega initial, \muk.

What is V_{cm} in terms of \omega?

While the ball is sliding, what is the ball's linear and angular acceleration?

How Long does the ball slide?

How Far does the ball slide?

What is the linear speed of the ball when smooth rolling begins?

Homework Equations



We know since it is sliding initially that the initial angular speed is zero

Rf_{s} = I_{cm}\alpha

Beyond this, I'm lost.
 
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The friction is a product of \mu_k and the weight (mg) or normal force to the horizontal surface.

The friction causes the ball to decelerate in terms of linear or translational motion while all causing the wall to increase in rotational velocity, i.e. the friction induces angular acceleration.

For moments of inertia, see - http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html, and
http://hyperphysics.phy-astr.gsu.edu/hbase/isph.html#sph

The ball stops sliding when Vcm = \omegar, i.e. the tangential speed at the radius = translational speed.
 
Kindly see the attached pdf. My attempt to solve it, is in it. I'm wondering if my solution is right. My idea is this: At any point of time, the ball may be assumed to be at an incline which is at an angle of θ(kindly see both the pics in the pdf file). The value of θ will continuously change and so will the value of friction. I'm not able to figure out, why my solution is wrong, if it is wrong .
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