A man opens a 1.14-m wide door by pushing on it with a force of 49.5 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 torque applied in each case, we need to use the formula:

Torque = Force * Distance * sin(θ)

Where:
- Torque is the magnitude of the torque applied (in N · m).
- Force is the magnitude of the force applied perpendicular to the door (in N).
- Distance is the shortest distance between the axis of rotation and the line of action of the force (in this case, the distance from the axis to the center or the edge of the door).
- θ is the angle between the force vector and the line connecting the axis and point of application of the force (in this case, 90 degrees).

Let's calculate the torque for each case:

(a) Torque applied at the center of the door:

Since the force is applied at the center of the door, the distance from the axis of rotation to the center is half the width of the door, which is 1.14 m / 2 = 0.57 m.

Torque = Force * Distance * sin(θ)
= 49.5 N * 0.57 m * sin(90 degrees)
= 49.5 N * 0.57 m * 1
= 28.215 N · m

Therefore, the magnitude of the torque applied about an axis through the hinges when the force is applied at the center of the door is 28.215 N · m.

(b) Torque applied at the edge farthest from the hinges:

Since the force is applied at the edge farthest from the hinges, the distance from the axis of rotation to this edge is half the width of the door, which is 1.14 m / 2 = 0.57 m.

Torque = Force * Distance * sin(θ)
= 49.5 N * 0.57 m * sin(90 degrees)
= 49.5 N * 0.57 m * 1
= 28.215 N · m

Therefore, the magnitude of the torque applied about an axis through the hinges when the force is applied at the edge farthest from the hinges is 28.215 N · m.