(a)
The total torques due to the weight of the hand about the axis of rotation when the time reads
(a)
Answer to Problem 40AP
The total torques due to the weight of the hand about the axis of rotation when the time reads
Explanation of Solution
Given information: The mass of hour hand is
Formula to calculate the net torque produced by the clock’s hand is,
Here,
Formula to calculate the angular speed of the hour hands is,
Substitute
Thus, the angular speed of the hour hand is
Formula to calculate the angular speed of the minute hands is,
Substitute
Thus, the angular speed of the hour hand is
Let take
Write the expression for the angular position of the hour hands at time
Here,
Substitute
Write the expression for the angular position of the minute hands at time
Here,
Substitute
Substitute
Substitute
When clock shows time
Substitute
Thus, the net torque is
When clock shows time
Substitute
Thus, the net torque is
When clock shows time
Substitute
Thus, the net torque is
When clock shows time
Substitute
Thus, the net torque is
When clock shows time
Substitute
Thus, the net torque is
Conclusion:
Therefore, the total torques due to the weight of the hand about the axis of rotation when the time reads
(b)
The all the time nearest to second when total torque about the axis of rotation is zero by solving the transcendental equation.
(b)
Answer to Problem 40AP
The time corresponding to the zero torque is given as:
Time(hr) | Clock time |
0 | 12:00:00 |
0.515 | 12:30:55 |
0.971 | 12:58:19 |
1.54 | 1:32:31 |
1.95 | 1:57:01 |
2.56 | 2:33:25 |
2.94 | 2:56:29 |
Explanation of Solution
Given information: The mass of hour hand is
From equation (2), the expression for the total torque is given by,
Substitute
Since it is a transcendental equation, solving the equation numerically the values of time comes out to be 0, 0.515, 0.971, 1.54, 1.95……so on.
The time corresponding to the time is given as:
Time(hr) | Clock time |
0 | 12:00:00 |
0.515 | 12:30:55 |
0.971 | 12:58:19 |
1.54 | 1:32:31 |
1.95 | 1:57:01 |
2.56 | 2:33:25 |
2.94 | 2:56:29 |
Conclusion:
Therefore, time corresponding to the zero torque is given as:
Time(hr) | Clock time |
0 | 12:00:00 |
0.515 | 12:30:55 |
0.971 | 12:58:19 |
1.54 | 1:32:31 |
1.95 | 1:57:01 |
2.56 | 2:33:25 |
2.94 | 2:56:29 |
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Chapter 10 Solutions
Physics for Scientists and Engineers with Modern Physics
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