0.32 Phy1320: angular momentum -- the mathematics of torque  (Page 13/13)

The torque will cause the door to rotate in a counter clockwise direction. Therefore, the torque has a positive sign.

Torque and the moment of inertia

Three objects are rotating about their centers. All three objects have a mass of 10 kg. The three objects have the followingshapes:

A. A solid disk with a moment of inertia given by

I = (1/2)*m*r^2

where

• r = 2m

B. A disk with a round hole in the center with a moment of inertia given by

I = (1/2)*m*(r1^2 + r2^2)

where

• r1 = 1m
• r2 = 2m

C. A square plate with a moment of inertia given by

I = (1/12)*m*(h^2 + w^2)

where

• h = w = 3.54m

Find the net torque required to cause each object to accelerate at a rate of 10 radians/sec^2.

Solution:

All three solutions are based on the general equation for torque given in Figure 5 .

A. T = I * A, or

T = (1/2)*m*r^2 * A, or

T = (1/2)*10kg*(2m)^2 * 10 radians/second^2

Entering this expression into the Google calculator gives us

T = 200 N*m

B. T = I * A, or

T = (1/2)*m*(r1^2 + r2^2) * A, or

T = (1/2)*10kg*((1m)^2 + (2m)^2) * 10 radians/second^2, or

T = 250 N*m

Note that because more of the mass is located close to the outer edge of the disk, the moment of inertia is higher and more torque is required to achieve thesame acceleration for the same mass.

C. T = I * A, or

T = (1/12)*m*(h^2 + w^2) * A, or

T = (1/12)*10kg*((3.54m)^2 + (3.54m)^2) * 10 radians/second^2, or

T = 2.09 N*m

Note that the square in part C was designed to have the same surface area as the disk in part A. The mass in both cases was uniformly distributed throughoutthe entire surface. Under those conditions, a square has a slightly higher moment of inertia than a disk and thus requires a slightly greater torque toachieve the same acceleration.

Repeat the computations

I encourage you to repeat the computations that I have presented in this lesson to confirm that you get the same results. Experiment withthe scenarios, making changes, and observing the results of your changes. Make certain that you can explain why your changes behave as they do.

Resources

I will publish a module containing consolidated links to resources on my Connexions web page and will update and add to the list as additional modulesin this collection are published.

Miscellaneous

This section contains a variety of miscellaneous information.

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