A brother (mass 20 kg) and his big sister (mass 32 kg) want to play on a teeter-totter. The beam is 7 m long and has a mass of 41 kg. Suppose the brother sits at the end of his side, 3.5 m away from the pivot point. How far away from the pivot point does the sister need to sit for the system to be in rotational equilibrium?

Use the center of mass fomula:

Xcm = (M1X1 + M2X2 + ... + MnXn)/(M1 + M2 + ... + Mn)
Don't forget the mass of the teeter-totter and its location at its geometric center (M1 = 41 kg, X1 = 3.5 since the fulcrum is in the middle of the beam).
Set the location of the sister at the origin (X = 0) and the brother at the opposite end of the beam (X = 7m).
Xcm = (20 x 7 + 32 x 0 + 41 x 3.5)/(20 + 32 + 41) = 283.5/93 = 3.05
The sister is 3.05 m from the pivot point.

To find the distance at which the sister needs to sit for the system to be in rotational equilibrium, we need to consider the torque exerted by each individual on the teeter-totter.

The torque can be calculated using the formula:

Torque = force × distance

Since we are dealing with rotational equilibrium, the sum of the torques on one side of the pivot point should be equal to the sum of the torques on the other side.

Let's start by calculating the torques exerted by the brother and the sister.

For the brother:
Torque_brother = force_brother × distance_brother

The force exerted by the brother is equivalent to his weight, which can be calculated as:
force_brother = mass_brother × gravitational acceleration

Substituting the known values:
force_brother = 20 kg × 9.8 m/s^2

Now, we can calculate the torque exerted by the brother using his force and distance from the pivot point:
Torque_brother = (20 kg × 9.8 m/s^2) × 3.5 m

Similarly, we can calculate the torque exerted by the sister.

For the sister:
Torque_sister = force_sister × distance_sister

Substituting the known values, we have:
force_sister = mass_sister × gravitational acceleration = 32 kg × 9.8 m/s^2

The distance for the sister is unknown, but it is the variable we are trying to find.

Now, we set up the equilibrium condition by equating the torques:
Torque_brother = Torque_sister

By substituting the respective torque formulas:
(20 kg × 9.8 m/s^2) × 3.5 m = (32 kg × 9.8 m/s^2) × distance_sister

Simplifying the equation:
(20 × 9.8) × 3.5 = (32 × 9.8) × distance_sister

Now, we can solve for the distance_sister:
distance_sister = (20 × 9.8 × 3.5) / (32 × 9.8)

Simplifying further:
distance_sister = 3.5 m

Therefore, the sister needs to sit 3.5 meters away from the pivot point for the system to be in rotational equilibrium.