A 46.4 -N force is applied to the outer edge of a door of width $1.26 \mathrm{m}$ in such a way that it acts (a) perpendicular to the door, (b) at an angle of \(43.0^{\circ}\) with respect to the door surface, (c) so that the line of action of the force passes through the axis of the door hinges. Find the torque for these three cases.

For this case, the force is acting perpendicular to the door. Therefore, the angle between the door surface and the force (θ) is 90º. Force, F = 46.4 N Distance (from hinge to outer edge), d = 1.26 m Angle, θ = 90° Now, let's calculate the torque using the formula: Torque = F × d × sinθ Torque = 46.4 × 1.26 × sin(90°) Torque = 58.464 Nm

In this case, the force is acting at a 43° angle with respect to the door surface. Therefore, the angle between the door surface and the force (θ) is 43º. Force, F = 46.4 N Distance (from hinge to outer edge), d = 1.26 m Angle, θ = 43° Now, let's calculate the torque using the formula: Torque = F × d × sinθ Torque = 46.4 × 1.26 × sin(43°) Torque ≈ 46.90 Nm

When the line of action of the force passes through the axis of rotation of the hinges, the angle between the force and the line joining the point at which force acts and the rotation axis (hinge) is 0º. Force, F = 46.4 N Distance (from hinge to outer edge), d = 1.26 m Angle, θ = 0° Now, let's calculate the torque using the formula: Torque = F × d × sinθ Torque = 46.4 × 1.26 × sin(0°) Torque = 0 Nm In this case, the torque is 0 Nm because the force is passing through the axis of rotation, causing no rotation of the door. To summarize: 1. The torque when the force is perpendicular to the door is 58.464 Nm. 2. The torque when the force is at a 43° angle with respect to the door surface is 46.90 Nm. 3. The torque when the force passes through the axis of the door hinges is 0 Nm.