The definition of torque is

 Where

 is the force

 is the distance

Theta is the angle between the direction of the force and the distance.

In this case, , so  .

Plugging in our remaining values:

Initial angular momentum is zero

Combine equations:

Solve for

Definition of :

Combine equations:

Plug in values:

The torque  is produced by a force  acting at a radius . Since the force and the radius are perpendicular, then the torque equation gives us:

The new torque, which we will call  is produced by a force of  acting at a radius of  at an angle of . Thus the torque equation gives us:

Since , plugging this in to the above gives us 

Torque is given by:

, where  is the length of the rod from the pivot point,  is the force acting on the rod, and  is the angle. Since we have all of these components, we can plug in and solve:

A system will only have a net torque of 0 when it has no net force or the net force goes through its center of gravity. The only forces applied in this system is from gravity. Therefore, we need to find the orientation at which all gravitational force goes through point p. This occurs when the rod is oriented vertically, thus . 

Note:  will also result in a net torque of 0. At this orientation, the rod is also vertical with the masses in swapped positions.

The equation for torque is . Looking at this equation we can infer that the maximum distance from the center would give maximum torque. Also, any angle besides  or  will give an absolute value of a number below one, giving us a smaller torque.

To calculate the torque on the pendulum, we need to know the position of the pendulum. We can find this using the following expression:

Note that we are using the cosine function because the pendulum begins at it's maximum angle. Plugging in our values:

The pendulum is still horizontal, but now on the other side. Now we can directly calculate the torque placed on the pendulum

Where the radius is the length of the pendulum and the force is the weight of the block (since the pendulum is horizontal).

For this question, we're given a number of scenarios in which an equal magnitude of force is applied to a rotating bar at a variety of different orientations and locations on the bar. We're then asked to identify which one would generate the most amount of torque.

First, let's recall that torque is a twisting force. That is, it is a force that causes an object to rotate about a pivot point, such as a sea-saw. We can write an expression for torque as follows.

Where  is the magnitude of the applied force,  is the distance of the applied force from the pivot point, and  is the angle between the applied force vector and the surface upon which the force is being applied.

Sometimes, the equation for torque is also expressed as follows.

Where  stands for the lever arm, which takes into account both  and . Thus, for any given magnitude of force, the torque will be the highest when  is greater and when  approaches .

With this expression in mind, we can look at each image and make a qualitative assessment of which one will have the greatest torque. We're looking for a diagram in which the force vector is furthest from the pivot point, and is also oriented as close to  with respect to the surface of the rotating object. This situation is described by the following picture, thus making it the correct answer.

Converting  to  and plugging in values

First, we must recall the formula for torque, which is 

 is the distance from the pivot, called the moment arm.  is the force, and  is the angle relative to the normal of the object.) Since all the forces are being applied perpendicular to the surface of the rod, . Thus,

The sum of the torques must equal zero, so David's torque plus Marc's torque must be the same as Paul's torque because David and Marc are applying torques in the opposite direction as Paul. This gives us 

 Dividing both sides by , we get Paul's force  to be