Solving Angular Momentum Problems with Linear & Angular Acceleration

[staf9]

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, $$\mu$$k.

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

R$$f_{s}$$ = $$I_{cm}$$$$\alpha$$

Beyond this, I'm lost.

 

[Astronuc]

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 = $\omega$r, i.e. the tangential speed at the radius = translational speed.

 

[ogb p]

I understand that angular momentum is a fundamental concept in classical physics and is often used to describe the rotational motion of objects. In this problem, we are dealing with a sphere that is initially sliding and then transitions to smooth rolling. To solve this problem, we can use principles of both linear and angular acceleration.

First, we can use the coefficient of kinetic friction and the known initial speed and angular speed to calculate the linear and angular acceleration of the ball. The linear acceleration can be found using the equation F = ma, where F is the frictional force and m is the mass of the ball. The angular acceleration can be found using the equation \alpha = \frac{F}{I_{cm}}, where I_{cm} is the moment of inertia of the ball.

Next, we can use the relationship between linear and angular acceleration, a = \alpha r, to find the radius of the ball's motion. This radius will also be the distance the ball travels while sliding before it transitions to smooth rolling.

To find the time it takes for the ball to slide, we can use the equation v = u + at, where v is the final speed (which is zero when the ball stops sliding), u is the initial speed (V_{cm}), and a is the linear acceleration we calculated earlier.

Finally, to find the linear speed of the ball when smooth rolling begins, we can use the equation v = \omega r, where \omega is the final angular speed (which we can calculate using the relationship \omega = \omega_{initial} + \alpha t) and r is the radius of the ball's motion.

In summary, to solve this problem, we need to use principles of both linear and angular acceleration, as well as the relationship between them, to calculate the various quantities such as linear and angular acceleration, distance, time, and final linear speed. I hope this explanation helps you understand the process of solving angular momentum problems.