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.
1)What is the force the left support exerts on the beam?
2)What is the force the right support exerts on the beam?
To find the force exerted by each support, we need to consider both the forces and torques acting on the beam.
1) To find the force exerted by the left support, let's consider the equilibrium of forces and torques on the beam. The beam is not rotating, so the sum of the torques about any point must be zero.
Firstly, consider the forces acting on the beam. The only vertical forces acting on the beam are the weight of the gymnast and the weight of the beam itself. The weight of the gymnast can be calculated using the formula F = m * g, where m is the mass of the gymnast (47 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
F_gymnast = m1 * g = 47 kg * 9.8 m/s^2
Next, let's consider the torques. The torque exerted by the weight of the gymnast is given by the equation Torque = Force * Distance. Since the gymnast is standing at the left end of the beam, the distance from the left support to the gymnast is L/3.
Torque_gymnast = F_gymnast * (L/3)
Now, to find the force exerted by the left support, we can set up an equation based on the equilibrium of torques:
Torque_left_support = Torque_gymnast + Torque_beam
Since the beam is not rotating, the torque exerted by the beam must be zero. Therefore:
Torque_left_support = Torque_gymnast
Now we can substitute in the values we have:
Torque_left_support = F_gymnast * (L/3)
Rearranging the equation to solve for the force exerted by the left support:
Force_left_support = Torque_left_support / (L/3)
2) Similarly, to find the force exerted by the right support, we can use the same approach. The only difference is that the distance from the right support to the gymnast is now 2L/3.
Torque_right_support = F_gymnast * (2L/3)
Similarly, solving for the force exerted by the right support:
Force_right_support = Torque_right_support / (L/3)
Now you can plug in the known values (m1, m2, and L) and calculate the forces exerted by the left and right supports.