A gymnast with mass m1 = 47 kg is on a balance beam that sits on (but is not attached to) two supports. The beam has a mass m2 = 118 kg and length L = 5 m. Each support is 1/3 of the way from each end. Initially the gymnast stands at the left end of the beam.

To find the position at which the beam balances, we can consider the torques acting on the beam due to the gymnast and the beam itself.

First, let's calculate the torques due to the gymnast. The torque can be defined as the product of the force and the perpendicular distance from the axis of rotation to the line of action of the force.

In this case, the torque due to the gymnast's weight acting on the left end of the beam can be calculated as follows:

Torque_gymnast = m1 * g * (L/3)

where m1 is the mass of the gymnast, g is the acceleration due to gravity, and L/3 is the distance from the left end of the beam to the support.

Next, we need to calculate the torque due to the beam itself. Since the beam is symmetrical, the weight of the beam can be considered to act at its center of mass, which is at the midpoint of the beam.

The torque due to the weight of the beam acting at its center of mass can be calculated as:

Torque_beam = m2 * g * (L/2)

where m2 is the mass of the beam, g is the acceleration due to gravity, and L/2 is the distance from the midpoint to the support.

Finally, the beam will balance when the sum of the torques due to the gymnast and the beam itself is zero.

Torque_gymnast + Torque_beam = 0

m1 * g * (L/3) + m2 * g * (L/2) = 0

Now we can substitute the given values into the equation and solve for the unknown:

47 kg * 9.8 m/s^2 * (5 m / 3) + 118 kg * 9.8 m/s^2 * (5 m / 2) = 0

After calculating this expression, we'll find the position at which the beam balances.