A koopaling encounters a stationary uniform spool of mass M = 69 slugs, outer ra

Question

A koopaling encounters a stationary uniform spool of mass M=69 slugs, outer radius R=4.5 in and inner radius r=3 in and with a radius of gyration about its longitudinal axis kzC=3.75 in. As part of its workout regimen it applies a constant force F = -3.1 i lb to the wound rope causing the spool to roll.

1)

What is the minimum friction coefficient urequired for rolling?

2. What is Roc(t')-Roc(0) i.e. the displacement of the spool's mass center C after a period of

t'=4s has elapsed?

HInts:

Be careful with units

Kinematics: What do you know about Ac where C is the spools mass center point?

Relate Ac to alpha where alpha is hte spools angular acceleration.

use the rolling condition. i.e. what do you know about velocity and acceleration of the conatact point.

Draw free body diagram

Conserve Linear momentum F=mac in X and Y directions

Conserve angular momentum in the Z direction. Recall that the point you sum moments about affects the equation you should use.

Solve the 3 equations for the 3 unknowns which will involve the normal force N and friction force Ff and perhaps the angular acceleration alpha

Explanation / Answer

Friction will act towards right.

1.

Torque = I*alpha

F*r - Fric*R = (mk^2)*(a/R)

3.1*3 - Fric*4.5 = (69*3.75^2)*a/4.5

Newton's second law: ma = Fric - F

69*a = Fric - 3.1

Solving both these eqns, a = 0.049 in/s^2 and Fric = 6.48 lb

Fric = u*M

u = 6.48 / (69*32.2) = 0.0029

2.

s = ut + 1/2*at^2

s = 0 + 1/2*0.049*4^2

s = 0.392 in

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