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.