Chapter 8, Problem 90AP

### College Physics

11th Edition
Raymond A. Serway + 1 other
ISBN: 9781305952300

Chapter
Section

### College Physics

11th Edition
Raymond A. Serway + 1 other
ISBN: 9781305952300
Textbook Problem

# A string is wrapped around a uniform cylinder of mass M and radius R. The cylinder is released from rest with the string vertical and its top end tied to a fixed bar (Fig. P8.90). Show that (a) the tension in the string is one-third the weight of the cylinder, (b) the magnitude of the acceleration of the center of gravity is 2g/3, and (c) the speed  of the center of gravity is (4gh/3)1/2 after the cylinder has descended through distance h. Verify your answer to part (c) with the energy approach.Figure P8.90

(a)

To determine
To verify the tension in the spring is one third of the weight of the cylinder.

Explanation

Given Info:

The mass of the cylinder is M; radius of the cylinder is R , h is the distance from the bar to centre of the cylinder.

The moving cylinder has both translational and rotational motion. Consider downward acceleration as negative.

Figure shows the tension on the string, weight of the cylinder, and direction of acceleration.

Apply translational effect on the cylinder. The cylinder will accelerate with an acceleration a.

From Newtonâ€™s second law,

Tâˆ’Mg=âˆ’Ma

• g is the acceleration due to gravity
• T is the tension in the string

Consider the inward torque as negative and outward torque as positive.

Formula to calculate torque on the cylinder is,

Ï„=âˆ’IÎ±

• Ï„ is the torque acting on the cylinder
• I is the moment of inertia of the cylinder
• Î± is the angular acceleration

Negative sign shows the direction on the torque.

Formula to calculate the moment of inertia I is,

I=MR22

Formula to calculate acceleration a in terms of Î± is,

Î±=aR

Use a/R for Î± and MR2/2 for I in Ï„=âˆ’IÎ± to rewrite Ï„

(b)

To determine
The magnitude of acceleration of the centre of gravity.

(c)

To determine
The magnitude of speed of the centre of gravity.

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