A man opens a 1.19-m wide door by pushing on it with a force of 54.5 N directed perpendicular to its surface.
(a) What magnitude torque does he apply about an axis through the hinges if the force is applied at the center of the door? ___N · m
(b) What magnitude torque does he apply at the edge farthest from the hinges? ___N · m
To find the magnitude of torque, we must first understand the concept of torque. Torque is defined as the product of the force applied and the distance from the axis of rotation to the line of action of the force. In this case, the axis of rotation is through the hinges, and the distance is the perpendicular distance from the axis to the line of action of the force.
(a) To find the torque when the force is applied at the center of the door, we need to determine the distance from the center of the door to the hinges. Since the door is 1.19 m wide, the distance from the center to the hinges is half the width, which is 1.19 m / 2 = 0.595 m.
The magnitude of torque is then given by the formula:
Torque = Force x Perpendicular Distance
In this case, the force is 54.5 N, and the perpendicular distance is 0.595 m.
Using the formula, we can calculate:
Torque = 54.5 N x 0.595 m = 32.4775 N · m
Therefore, the magnitude torque applied about the axis through the hinges when the force is applied at the center of the door is 32.4775 N · m.
(b) To find the torque at the edge farthest from the hinges, we need to determine the distance from the hinges to the edge of the door. Since the door is 1.19 m wide, the distance from the hinges to the edge farthest from the hinges is half the width, which is 1.19 m / 2 = 0.595 m.
The magnitude of torque is then given by the formula:
Torque = Force x Perpendicular Distance
In this case, the force is 54.5 N, and the perpendicular distance is 0.595 m.
Using the formula, we can calculate:
Torque = 54.5 N x 0.595 m = 32.4775 N · m
Therefore, the magnitude torque applied at the edge farthest from the hinges is also 32.4775 N · m.
To find the torque applied by the man, we can use the formula:
Torque = Force * Distance * sin(theta)
(a) When the force is applied at the center of the door, the distance from the center to the hinges is half the width of the door, so Distance = 1.19 m / 2 = 0.595 m. The angle theta between the force and the lever arm is 90 degrees.
So, Torque = 54.5 N * 0.595 m * sin(90 degrees) = 32.4275 N · m (rounded to four decimal places)
Therefore, the magnitude torque applied about an axis through the hinges when the force is applied at the center of the door is approximately 32.4275 N · m.
(b) When the force is applied at the edge farthest from the hinges, the distance from the edge to the hinges is half the width of the door, so Distance = 1.19 m / 2 = 0.595 m. The angle theta between the force and the lever arm is 90 degrees.
So, Torque = 54.5 N * 0.595 m * sin(90 degrees) = 32.4275 N · m (rounded to four decimal places)
Therefore, the magnitude torque applied at the edge farthest from the hinges is approximately 32.4275 N · m.