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Physics: Bicycle mass of rim, angular velocity of wheel, rotations, work done; Cylinder, moment of inertia, center, speed, height, omega

A bicycle wheel consist of a rim ( radius R = 0. 40 m ) and essintially massless spokes. Soppose we mount the bycle wheel on the wall. so that the wheel is parallel to the wall. Is it free to spin around its center, but not freee to translate. A very light string is wrapped around the rim and a mass m + 0. 80 kg is attached to the end of the string. The mass is then released and falls 1.5 m in 0.78 s

Determine the mass of the rim.
Determine the final angular velocity of the wheel
Through how many rotations did the wheel spin?
Calculate the work done by the mass on the wheel

A solid cylinder of mass M and raduis R is rolling without slipping towards a semicircular track. The cylinder's rotation velocity around its central axis is omega 0 The track is attached to the floor. The radius of the semicircle is a. Note that a>>R. the cylinder roll up the track without slipping until it looses contact with the track at point P.

Derive the moment of inertia for the cylinder about itel center axis
Determine the cylinder's center - of - madd speed when it reaches point Q
Determine the height of point P in term of g, R , a, and omega 0

Solution Preview

See attached files.

Solution and explanation:

FIRST PROBLEM

Step 1.
Starting from rest, the mass m= .8 kg descends distance h= 1.5 m at constant acceleration a, in time t= .78 sec. The motion equation which relates these parameters is:
(1) a = 2 h / t^2
Substituting knowns into (1) you should get:
(2) a = 4.93 m/sec^2

Step 2.
Linear acceleration a is related to angular acceleration A by:
(3) a = R A
By solving for A and substitution of knowns you should get:
(4) A = 12.3 rad/sec

Step 3.
The mass of the rim involves moment of inertia of the wheel, (a thin ring), which is:
(5) I = M R^2 about an axis at its center of mass.

Step 4.
The cord force C is determined by applying 'net force = m a' to the descending mass. The cord pulls up on mass m and force mg is downward. Thus:
(6) m g - C = m a

Step 5.
Substituting knowns and solving (6) for C should ...

Solution Summary

The solutions are clearly explained and solved using steps to work to solutions.

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