A crank .25 long turns a shaft with a diameter of 3cm around which part of a rope is wound. One end of the rope hongs free and supports a weight of 4200nt. (a) What torque does the weight exert on the shaft? (b) What torque must be excerted on the crank to keep the system in equilibrium? (c) What force must be exerted to turn the crank?
To solve this problem, we can use the basic formula for torque:
Torque = Force * Distance (from the axis of rotation)
(a) To find the torque exerted by the weight, we need to know the distance from the axis of rotation (shaft) to the point where the rope is wound. Since the diameter of the shaft is 3 cm, the radius would be 1.5 cm or 0.015 m. The length of the crank is given as 0.25 m.
The distance from the axis of rotation to the point where the rope is wound can be calculated as follows:
Distance = crank length - radius of the shaft
Distance = 0.25 m - 0.015 m
Distance = 0.235 m
Now, we can calculate the torque exerted by the weight:
Torque = Force * Distance
Torque = 4200 N * 0.235 m
Torque = 987 Nm
Therefore, the torque exerted by the weight on the shaft is 987 Nm.
(b) In equilibrium, the total torque exerted on the system must be zero. The weight exerts a torque of 987 Nm in one direction, so there must be an opposing torque in the opposite direction to balance the system.
To find the torque required to keep the system in equilibrium:
Total torque = Torque exerted by the weight
Total torque = -987 Nm (negative sign indicates opposite direction)
Therefore, the torque that must be exerted on the crank to keep the system in equilibrium is -987 Nm.
(c) To find the force that must be exerted to turn the crank, we need to know the distance from the axis of rotation (shaft) to the point where the force is applied. This distance is the same as the crank length.
Force = Torque / Distance
Force = 987 Nm / 0.25 m
Force = 3948 N
Therefore, the force that must be exerted to turn the crank is approximately 3948 N.
To answer these questions, we need to understand the concept of torque and how it relates to rotational motion. Torque is the measure of the ability of a force to cause rotation around an axis or a point. It depends on two factors: the magnitude of the force applied and the perpendicular distance between the force and the axis of rotation. Mathematically, torque (τ) is given by the equation: torque = force × distance.
Let's break down each question to find the answers:
(a) What torque does the weight exert on the shaft?
In this case, the weight is hanging freely on one end of the rope. Since the weight is 4200 N, it will exert a force of 4200 N downwards. To find the torque, we need to determine the perpendicular distance between the force and the axis of rotation.
The rope is wound around the shaft, so the force is acting tangentially to the shaft. We can calculate the circumference of the shaft using its diameter, which is 3 cm (or 0.03 m): circumference = π × diameter = π × 0.03 m.
Since the rope is wound for 0.25 long turns, we multiply the circumference by 0.25 to find the distance: distance = 0.25 × circumference.
Now we have the force (4200 N) and the distance (0.25 × circumference). We can calculate the torque exerted by the weight using the equation: torque = force × distance.
(b) What torque must be exerted on the crank to keep the system in equilibrium?
In this case, the system is in equilibrium, which means the total torque exerted on the crank must be zero. The torque exerted by the weight (calculated in part (a)) will be in the opposite direction of the torque applied to keep the system in equilibrium.
To find the torque required to balance the weight, we need to know the perpendicular distance between the force applied to the crank and the axis of rotation. However, this information is not provided in the question. Without the perpendicular distance, we cannot determine the torque needed to keep the system in equilibrium.
(c) What force must be exerted to turn the crank?
To find the force needed to turn the crank, we first need to determine the torque required. However, we are missing crucial information in the question, such as the perpendicular distance between the force applied to the crank and the axis of rotation. Without this distance, we cannot calculate the force required to turn the crank accurately.
In summary:
(a) To find the torque exerted by the weight on the shaft, use the equation τ = force × distance, where the force is 4200 N (weight) and the distance is 0.25 × circumference of the shaft.
(b) The torque required to keep the system in equilibrium cannot be determined with the information provided.
(c) The force required to turn the crank cannot be determined accurately without the missing perpendicular distance.