A force is applied to a door at an angle of 56.8° and 0.34 m from the hinge.

a) What force produces a torque with a magnitude of 2.18 N·m?

b) How large is the maximum torque this force can exert?

Fdsin theda=T

(sin 56.8*)(.34)F=2.18
F=7.66

To solve these problems, we will use the formula for torque:

Torque = Force * Distance * sin(angle)

Given:

Angle (θ) = 56.8°
Distance (r) = 0.34 m
Torque (τ) = 2.18 N·m

a) To find the force that produces a torque of 2.18 N·m:

Rearranging the formula, we have:

Force = Torque / (Distance * sin(angle))

Substituting the given values:

Force = 2.18 N·m / (0.34 m * sin(56.8°))

Calculating the force:

Force = 2.18 N·m / (0.34 m * 0.837)

Force ≈ 7.16 N

Therefore, a force of approximately 7.16 N produces a torque of 2.18 N·m.

b) To find the maximum torque this force can exert:

The maximum torque occurs when the force is applied perpendicular to the door, which happens when the angle is 90°. In this case, sin(90°) = 1.

Torque = Force * Distance * sin(angle)

Substituting the maximum values:

Max Torque = Force * Distance * sin(90°)

Max Torque = Force * Distance

Max Torque = 7.16 N * 0.34 m

Max Torque ≈ 2.44 N·m

Therefore, the maximum torque this force can exert is approximately 2.44 N·m.

To solve these problems, we will use the concept of torque and the formula of torque.

Torque (τ) is the product of force (F) and the perpendicular distance (r) between the line of action of the force and the axis of rotation (hinge in this case). Mathematically, it can be expressed as:

τ = F * r * sin(θ),

where θ is the angle between the force and the perpendicular distance.

Let's solve each part of the problem:

a) What force produces a torque with a magnitude of 2.18 N·m?

We know the magnitude of torque (τ) is 2.18 N·m, the angle (θ) is 56.8°, and the distance (r) is 0.34 m.

Using the formula, we have:

τ = F * r * sin(θ)

Substituting the known values:

2.18 N·m = F * 0.34 m * sin(56.8°)

Now, we can isolate the force (F):

F = 2.18 N·m / (0.34 m * sin(56.8°))

Calculating this expression will give us the force (F).

b) How large is the maximum torque this force can exert?

To determine the maximum torque, we need to consider the angle (θ) that will produce the maximum torque.

Since torque is maximum when the force is applied perpendicular to the distance, the angle (θ) will be 90°.

Using the same formula:

τ = F * r * sin(θ)

Maximum torque (τ_max) will be:

τ_max = F * r * sin(90°)

sin(90°) = 1, so the expression simplifies to:

τ_max = F * r

Plugging in the known values, force (F) and distance (r), will give us the maximum torque (τ_max).