Question

A thin uniform disk of mass m and radius r has a string wrapped

Answer 1

The magnitude of the weight of the disk is greater than the magnitude of the tension force of the string. ((c) $F_{T} < F_{g}$)

How to analysis a rigid body on dynamical condition

A rigid body is a system whose geometry cannot be neglected and in which deformations due to external forces applied on the system are negligible.

In this question we need to establish a free body diagram with all critical forces and motion conditions. There are two external forces: (i) tension ($F_{T}$), (ii) weight ($F_{g}$), both in newtons.

The disk falls due to gravity and rotates around the string in a case of rolling, which combines rotation and translation. By Newton's laws of motion we have the following expression:

$\sum F = -F_{g}+F_{T} = -m\cdot R\cdot \alpha$   (1)

Where:

• $m$ - Mass of the disk, in kilograms.
• $\alpha$ - Angular acceleration, in radians per square second.

By algebra we derive the following relationship between the two forces:

$-F_{g} = -F_{T}-m\cdot R\cdot \alpha$

$F_{g} > F_{T}$

Therefore, the magnitude of the weight of the disk is greater than the magnitude of the tension force of the string. ((c) $F_{T} < F_{g}$) $\blacksquare$

Remark

The statement is incomplete and poorly formatted. Correct and complete form is shown below:

A thin uniform disk of mass m and radius r has a string wrapped around its edge and attached to the ceiling. The bottom of the disk is at height 3R above the floor, as shown above. The disk is released from rest. The rotational inertia of a disk around its center is $I = \frac{1}{2}\cdot M\cdot R^{2}$.

When released from rest, the disk falls and the string unwinds. The force that the string exerts on the disk is $F_{T}$, and the gravitational force exerted on the disk is $F_{g}$. Which of the following expression correctly relates $F_{T}$ and $F_{g}$ as the disk falls?

(a) $F_{T} > F_{g}$, (b) $F_{T} = F_{g}$, (c) $F_{T} < F_{g}$

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