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## Homework Statement

A uniform sphere with know mass m, moment of inertia I, and radius R

_{s}is traveling through free space with initial horizontal linear velocity v

_{1}and rotational velocity w

_{1}. It then makes tangential contact with a horizontal surface and instantaneously starts rolling without slipping. What is the rotation/linear velocity the instant after it starts rolling?

Assume positive linear velocity is to the left, and positive rotation is counter-clockwise.

EDIT: assume a gravitational force sufficient enough to induce a frictional force to cause no sliding.

## Homework Equations

F

_{x}= m*a

_{x}

a

_{x}= (v

_{2}-v

_{1})/t

F*R = I*alpha

alpha = (w

_{2}-w

_{1})/t

impulse = F*t

*during no slip rolling*

v

_{x}= w*R

_{s}

## The Attempt at a Solution

I would try to solve this problem by trying to apply a horizontal impulse to the sphere, changing the linear velocity and the angular velocity, until the new linear velocity matched the linear velocity induced by rolling (v

_{x}= w

_{s}*R

_{s}) therefore:

(1) F = m*a

_{x}=> (F*t)/m = v

_{2}-v

_{1}

(2) F*R

_{s}= I

_{rolling}*alpha => (

**F**

(3) v

This is 3 equations and three unknows (v[/B]

***R*****t)**_{s}/I_{rolling}= w_{2}-w_{1}(3) v

_{2}= w_{2}*R_{s}This is 3 equations and three unknows (v

_{2}, w_{2},**F*t). Solving this we get:**

*only here for you to check my math, actual reading not necessary**only here for you to check my math, actual reading not necessary*

**substituting**

**(3) into (1):**

**(F*t)/m =**

My main question is: is this the correct way to attack the problem? (apply a impulse until the linear and rotational speeds match). This concept will later be applied to a much more complex system (dynamics simulation). Is it still valid to apply this to a system with complex internal forces, but who's total mass is m, total linear velocity is v**-v****w**_{2}*R_{s}_{1}**=>****(F*t) = (****-v****w**_{2}*R_{s}_{1})*m (equation: 4)**substituting****(4) into (2):**

**(w**_{2}*R_{s}-v_{1}***R****)*m**_{s}/I_{rolling}= w_{2}-w_{1}=> and I'm not going to finish typing that because it takes a long time and the equation formatter is buggy. I realize that that I need to do a substitution of I_{rolling}= I + m*R_{s}^{2}(parallel axis theorem because while the sphere is on the ground it is rotating about its edge, not its center of mass)My main question is: is this the correct way to attack the problem? (apply a impulse until the linear and rotational speeds match). This concept will later be applied to a much more complex system (dynamics simulation). Is it still valid to apply this to a system with complex internal forces, but who's total mass is m, total linear velocity is v

_{1}, effective radius is R, and total moment of inertia is I?
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