The length of a bicycle pedal arm is 0.152 m, and a downward force of 146 N is applied to the pedal by the rider's foot. What is the magnitude of the torque about the pedal arm's pivot point when the arm makes the following angles with the vertical?
(a) 30°
(b) 90°
(c) 180°
To find the magnitude of the torque about the pedal arm's pivot point, we can use the formula:
Torque = Force * lever arm
where:
- Torque is the torque about the pivot point,
- Force is the force applied to the pedal by the rider's foot, and
- Lever arm is the distance from the pivot point to the line of action of the force.
For each angle, we need to calculate the lever arm and then use the formula to find the torque.
(a) When the arm makes an angle of 30° with the vertical:
The lever arm is the horizontal distance between the pivot point and the line of action of the force. This can be found using trigonometry:
lever arm = length of pedal arm * sin(angle)
lever arm = 0.152 m * sin(30°) ≈ 0.076 m
Now we can calculate the torque:
Torque = Force * lever arm
Torque = 146 N * 0.076 m ≈ 11.096 N·m
Therefore, the magnitude of the torque about the pedal arm's pivot point at an angle of 30° is approximately 11.096 N·m.
(b) When the arm makes an angle of 90° with the vertical:
In this case, the lever arm is equal to the length of the pedal arm:
lever arm = 0.152 m
Now we can calculate the torque:
Torque = Force * lever arm
Torque = 146 N * 0.152 m ≈ 22.192 N·m
Therefore, the magnitude of the torque about the pedal arm's pivot point at an angle of 90° is approximately 22.192 N·m.
(c) When the arm makes an angle of 180° with the vertical:
In this case, the lever arm is zero because the line of action of the force coincides with the pivot point.
Now we can calculate the torque:
Torque = Force * lever arm
Torque = 146 N * 0 = 0 N·m
Therefore, the magnitude of the torque about the pedal arm's pivot point at an angle of 180° is zero.
To summarize:
- Torque at 30° = approximately 11.096 N·m
- Torque at 90° = approximately 22.192 N·m
- Torque at 180° = 0 N·m