Understanding Rotational Motion: Solving for Linear and Angular Acceleration

In summary, the problem involves a sphere placed on a table with a coefficient of kinetic friction μ. The sphere has an initial velocity of center of mass Vo = 0 and is rotating with angular velocity ωo around a horizontal axis. The goal is to find the linear acceleration of the center of mass and angular acceleration of the sphere. The attached picture shows the setup of the problem. The main confusion is how the sphere can spin but not move linearly, which is due to skidding on the surface.
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Homework Statement



Consider a sphere that is placed at a table. The coefficient of kinetic friction between the sphere and the table is μ. At time t = 0, the velocity of the center of mass of the sphere is Vo = 0, and the sphere is rotating with respect of the horizontal axis with angular velocity ωo.
Find the linear acceleration of the center of mass of the sphere and the angular acceleration of sphere. (picture attached)

https://www.physicsforums.com/attachment.php?attachmentid=57185&d=1364426694

Homework Equations

The Attempt at a Solution



So I'm having a hard time understanding rotation problem. Can anyone please explain this to me and the approach to solve it? Thank you
 

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What in particular are you having trouble understanding? The sphere is skidding on the surface if you're wondering how it can spin yet not move linearly.
 

What is rotational motion?

Rotational motion is the movement of an object around a fixed axis. It is characterized by the rotation of the object, rather than its linear movement. Examples of rotational motion include the spinning of a top or the rotation of the Earth around its axis.

What causes rotational motion?

Rotational motion is caused by a force acting on an object that is not directly on its center of mass. This creates a torque, which is the force that causes rotational motion. Torque is calculated by multiplying the force by the distance from the axis of rotation to the point where the force is applied.

What is angular velocity?

Angular velocity is the rate at which an object rotates around an axis. It is measured in radians per second and is calculated by dividing the change in angle by the change in time. The direction of angular velocity is determined by the right-hand rule, where the fingers of the right hand curl in the direction of rotation.

What is moment of inertia?

Moment of inertia is a measure of an object's resistance to rotational motion. It is similar to mass in linear motion, and it is calculated by summing the products of each particle's mass and its square of distance from the axis of rotation. In simpler terms, it represents how spread out the mass of an object is from its axis of rotation.

How does rotational motion relate to linear motion?

Rotational motion and linear motion are closely related and can be converted back and forth using the equations of kinematics. For example, angular displacement is equivalent to linear displacement divided by the radius, and angular velocity is equivalent to linear velocity divided by the radius. Additionally, forces and energy can be transferred between rotational and linear motion systems.

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