The right-hand 87 𝑁 force is applied perfectly vertical (+y) at a distance 𝑑1 = 0.32 𝑚 from the axis of rotation of the wrench at an angle of 39 degrees. If the same force were to be applied at a distance 𝑑2 = 0.28 𝑚 from the axis instead (the left-hand force as shown), at what angle θ would this force need to be applied to produce the same torque or moment about the axis of rotation

To solve this problem, we need to equate the torque produced by the right-hand force at distance \( d_1 \) with the torque produced by the left-hand force at distance \( d_2 \), and find the required angle \( \theta \) for the left-hand force.

### Step 1: Torque formula
The torque \( \tau \) produced by a force \( F \) applied at a distance \( d \) from an axis of rotation is given by:
\[
\tau = F \cdot d \cdot \sin(\phi)
\]
where \( \phi \) is the angle between the direction of the force and the lever arm (distance from the axis).

### Step 2: Torque produced by the right-hand force
For the right-hand force:
- Force \( F = 87 \, \text{N} \),
- Distance \( d_1 = 0.32 \, \text{m} \),
- Angle \( \phi_1 = 39^\circ \).

The torque produced by the right-hand force is:
\[
\tau_1 = 87 \cdot 0.32 \cdot \sin(39^\circ)
\]

### Step 3: Torque produced by the left-hand force
For the left-hand force:
- Force \( F = 87 \, \text{N} \),
- Distance \( d_2 = 0.28 \, \text{m} \),
- Angle \( \theta \) (unknown).

The torque produced by the left-hand force is:
\[
\tau_2 = 87 \cdot 0.28 \cdot \sin(\theta)
\]

### Step 4: Equating the torques
To produce the same torque, \( \tau_1 = \tau_2 \). Thus:
\[
87 \cdot 0.32 \cdot \sin(39^\circ) = 87 \cdot 0.28 \cdot \sin(\theta)
\]

We can cancel the common factor \( 87 \) from both sides:
\[
0.32 \cdot \sin(39^\circ) = 0.28 \cdot \sin(\theta)
\]

### Step 5: Solving for \( \sin(\theta) \)
Now solve for \( \sin(\theta) \):
\[
\sin(\theta) = \frac{0.32 \cdot \sin(39^\circ)}{0.28}
\]

Let's calculate this value.

### Step 6: Calculation
Now I'll compute the value of \( \theta \).The angle \( \theta \) at which the left-hand force should be applied to produce the same torque is approximately \( 46^\circ \).

Would you like further details or have any questions?

Here are 5 related questions you might explore:
1. How does torque change if the force is applied at a different angle but with the same magnitude?
2. What happens to the torque if the distance from the axis is doubled?
3. How is torque related to rotational equilibrium in a system?
4. How would the problem change if the force were applied at a downward angle?
5. What is the effect of applying the force exactly perpendicular to the lever arm?

**Tip:** Always check whether the force direction is perpendicular to the lever arm for maximum torque efficiency.

The right-hand 87 N force is applied perfectly vertical (+y) at a distance d1 = 0.32 m from the axis of rotation of the wrench at an angle of 39 degrees. If the same force were to be applied at a distance d2 = 0.28 m from the axis instead (the left-hand force as shown), at what angle θ would this force need to be applied to produce the same torque or moment about the axis of rotation?

Discover how to calculate the angle needed to apply a force of 87 N at different distances to maintain the same torque on a wrench. This problem involves torque calculations using trigonometric functions and rotational mechanics. Suitable for high school students in Grades 10-12.

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