Question

two forces are applied to a ships wheel. the wheel is not rotating. which of the following must be true?

the magnitude of the torque created by ( f_1 ) must be equal in magnitude to the torque created by ( f_2 ).
the magnitude of the torque created by ( f_1 ) must be greater than the magnitude of the torque created by ( f_2 ).
the magnitude of the torque created by ( f_2 ) must be greater than the magnitude of the torque created by ( f_1 ).
( f_1 ) must be equal to ( f_2 ).

Explanation:

Step1: Recall the condition for rotational equilibrium

When an object is not rotating (in rotational equilibrium), the net torque acting on it is zero. Torque is given by $\tau = rF\sin\theta$ (where $r$ is the distance from the axis of rotation to the point of force application, $F$ is the force magnitude, and $\theta$ is the angle between the force vector and the position vector from the axis to the point of force application).

Step2: Analyze the given situation

For the ship's wheel not to rotate, the sum of torques $\tau_1+\tau_2 = 0$. This implies $\tau_1=-\tau_2$, or in terms of magnitudes $|\tau_1| = |\tau_2|$. The equality of forces ($F_1 = F_2$) is not necessarily required because the torques also depend on $r$ (the lever - arm) and $\theta$ (the angle of force application). For example, if $F_1$ acts at a larger $r$ (lever - arm) than $F_2$, $F_1$ could be smaller in magnitude than $F_2$ and still create the same torque.

Answer:

The magnitude of the torque created by $F_1$ must be equal in magnitude to the torque created by $F_2$.