The formula for the potential energy in a spring is:

Use the given spring constant and displacement to solve for the stored energy.

If no outside forces act upon the pendulum, it will continue to oscillate back to the original height of . 

The proof of this is in the law of conservation of energy. At the top, the pendulum has all potential energy, which is given by the formula . As it swings, the potential energy is converted to kinetic energy until, at the bottommost point, there is only kinetic energy. It then changes direction and begins to rise again. When it rises to the maximum height on the other side, all of its kinetic energy will turn back into potential energy.

Mathematically, the initial and final potential energies are equal.

Notice the masses and gravity can cancel out on both sides, as neither of these will change. This leaves us with only height.

This is a static equilibrium problem.  In order for static equilibrium to be achieved, there are three things that must be true.  First, the sum of the forces in the horizontal direction must all equal .  Second, the sum of the forces in the vertical direction must all be equal to zero.  Third, the torque around a fixed axis must equal .

Let us begin by summing up the forces in the vertical direction.

Then let us sum up the forces in the horizontal direction

Lastly let us analyze the torque, using the hinge as the axis point where  is the length of the beam.

Looking at these three equations, the easiest to work with would be our torque equation as it is only missing one variable.  If we are able to find the tension in the  direction we would then be able to use trigonometry to determine the tension in the cable overall.

We can now substitute this value back into our vertical direction equation to determine the force on the hinge in the  direction.

We can go back to our tension in our  direction and use trigonometry to determine the force of the tension in  direction.

Rearrange and solve for the tension in the  direction.

We can now go back to our second equation and substitute this value in.

First let us analyze the torque that is happening on the bucket.  There is a torque from the friction on the pulley and there is a torque from the bucket pulling on the pulley.

Next let us analyze the forces involved.  There is a tension force pulling up on the bucket and there is the force of gravity pulling down on the bucket.

We can rearrange this to find  so that we can substitute it into our torque equation.

Now substitute this into our torque equation

We know that there is a relationship between acceleration and angular acceleration.

So we can substitute this into our equation so that only linear acceleration is present.

We also know that the moment of inertia of the system is equal to the moment of inertia of the pulley plus the bucket.

We can now substitute this into our equation.

We can now starting putting in our known variables to solve for the missing acceleration.

The formula for torque is:

We are given the total torque and the force applied. Using these values, we can solve for the length of the wrench.

If the see-saw is  in total, then it has  on either side of the fulcrum. 

The question is asking us to find the equilibrium point; that means we want the net torque to equal zero.

Now, find the torque for the first child.

We are going to use the force of gravity for the force of the child.

When thinking of torque, treat the positive/negative as being clockwise vs. counter-clockwise instead of up vs. down. In this case, child one is generating counter-clockwise torque. That means that since , child two will be generating clockwise torque.

Solve for the radius (distance) of the second child.

Torque is equal to the force applied perpendicular to a surface, multiplied by the radius or distance to the pivot point.

In this case, one force is applied perpendicular and the other at an angle.  The one that is applied at an angle, only has a small component of the total force acting in the perpendicular direction.  This component will be smaller than the overall force.  Therefore the force that is already acting perpendicular to door will provide the greatest torque.

Torque is equal to the force applied perpendicular to a surface, multiplied by the radius or distance to the pivot point.

In this example the boy of mass M is a distance R away and is balancing a girl of mass m at a distance r away.

If both of these kids move to a distance that is one half their original distance.

The half cancels out of the equation and therefore the boy and girl will still be balanced.

Torque is equal to the force applied perpendicular to a surface, multiplied by the radius or distance to the pivot point.

In this case, both forces are equal to one another.  Therefore the force that is applied at the point furthest from the axis of rotation (the hinge) will have the greater torque.  In this case, the furthest distance is the doorknob.

Torque is a force times the radius of the circle, given by the formula:

In this case, we are given the radius in centimeters, so be sure to convert to meters:

Use this radius and the given force to solve for the torque.