Rotational kinematics (mass wrapped around inner hub)

In summary, a bicycle wheel with a mass of 6.55 kg, radius of 38.0 cm and a ring shape is set up in a lab with a mass of 1.85 kg attached to a string wrapped around an inner hub with a radius of 5.40 cm. The mass is initially 72.0 cm above the floor and friction is negligible. The resulting angular acceleration of the wheel is 1.06 rad/s^2, it takes 0.71 seconds for the mass to reach the floor, the total angular displacement of the wheel during this time is 0.378 radians, and the work done on the wheel by external torque is 2.57 J. The calculations were done
  • #1
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Homework Statement


A bicycle wheel is mounted as in the lab and as shown to the right. This wheel has a mass of
6.55 kg, a radius of R = 38.0 cm and is in the shape of a ring. A mass M = 1.85 kg is attached to
the end of a string which is wrapped around an inner hub which has a radius r = 5.40 cm.

Initially, the mass M is a distance h = 72.0 cm above the floor. [Assume friction is negligible!]
a. What will be the resulting angular acceleration of this wheel?
b. How long will it take for the mass M to reach the floor?
c. What will be the total angular displacement of the wheel during the time in which the mass M
is falling to the floor?
d. How much work was done on the wheel by the external torque as the mass M falls to the floor?

Homework Equations


torque = FR = I * alpha
F = Ma
a = alpha * r

I = mr^2 (told to ignore the spokes of wheel, same as hoop)

The Attempt at a Solution


I tried using T*R = I * alpha (T = tension, I = mR^2 where m is mass of wheel)
and Mg - Ma = T and a = alpha *r

I substituted for T, combined them together to solve for alpha,
got alpha = MgR/(mR^2 + MR^2)
which is 5.795

however answer is 1.06rad/s^2. Can you help?
 
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  • #2
T*R = I*alpha? If I understand the problem description correctly, the tension T acts at a distance r from the hub, not R.

That would explain why the answer you derived is independent of r while the real answer should depend strongly on r.
 
  • #3
that did it thanks! Mgr/(MR^2 + Mr^2) got me the answer
 

1. What is rotational kinematics?

Rotational kinematics is the study of the motion of objects that are rotating around an axis. It involves concepts such as angular velocity, angular acceleration, and rotational inertia.

2. How is rotational kinematics different from linear kinematics?

Rotational kinematics deals with the motion of objects that are rotating, while linear kinematics deals with the motion of objects in a straight line. Rotational kinematics also involves different equations and concepts, such as torque and moment of inertia.

3. What is the equation for linear speed in rotational kinematics?

The equation for linear speed in rotational kinematics is v = ωr, where v is the linear speed, ω is the angular velocity, and r is the distance from the axis of rotation.

4. How does mass affect rotational kinematics?

Mass does not directly affect rotational kinematics, but it does affect the rotational inertia of an object. The greater the mass of an object, the greater its resistance to changes in rotational motion.

5. What is the difference between rotational velocity and angular velocity?

Rotational velocity is the speed at which an object is rotating around an axis, while angular velocity is the rate of change of the angular displacement of an object. Rotational velocity is measured in radians per second, while angular velocity is measured in degrees per second or revolutions per second.

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