A crank .25m long turns a shaft with a diameter of 3cm around which part of a rope is wound. One end of the rope hangs free and supports a weight of 4200nt. (a) What torque does the weight exert on the shaft? (b) WHat torque must be exerted on the crank to keep the system in equilibrium? (c) What force must be exerted to turn the crank?
To find the torque exerted on the shaft by the weight, we can first calculate the force exerted by the weight and then multiply it by the perpendicular distance from the axis of rotation (shaft) to the line of action of force.
(a) Torque exerted by the weight:
- First, calculate the force exerted by the weight using Newton's second law, F = m * g, where m is the mass and g is the acceleration due to gravity (9.8 m/s^2).
- The weight is given as 4200 N, and since weight = force, the mass (m) can be calculated as 4200 N / 9.8 m/s^2.
- Now, we substitute the calculated mass into the formula to find the force.
- Once we have the force, we calculate the torque by multiplying it by the perpendicular distance from the axis of rotation to the line of action of force. The distance is given as the radius of the shaft, which is half the diameter (3 cm / 2).
(b) To keep the system in equilibrium, the torque exerted on the crank must balance the torque exerted by the weight.
- Since the crank is rotating, we only consider the torque exerted by the weight on the opposite side of the shaft.
- The torque exerted by the weight on the crank is equal in magnitude but opposite in direction to the torque exerted by the weight on the shaft.
(c) To turn the crank, a force must be exerted at a distance from the axis of rotation.
- We can calculate the force needed to turn the crank by using the torque required and dividing it by the distance from the axis of rotation.
Note: To calculate torque, we assume the force and distance are perpendicular to each other.
Let's calculate each part step by step.
(a) Torque exerted by the weight:
- Mass (m) = 4200 N / 9.8 m/s^2 = 428.57 kg (approx.)
- Force (F) = 4200 N
- Radius of the shaft (r) = 3 cm / 2 = 1.5 cm = 0.015 m
Torque (τ) = F * r
= 4200 N * 0.015 m
= 63 Nm
The torque exerted by the weight on the shaft is 63 Nm.
(b) Torque exerted on the crank to keep the system in equilibrium:
Since the crank is rotating, we consider its opposite side.
The torque exerted on the crank is equal in magnitude but opposite in direction to the torque exerted on the shaft.
Therefore, the torque is also 63 Nm.
(c) Force required to turn the crank:
To find the force, we need to know the distance from the axis of rotation.
If the distance is given, we can calculate the force using the formula:
Force (F) = Torque (τ) / Distance (r)
Please provide the distance from the axis of rotation, and I can assist you further with the calculation.