A 3m long balance beam is simply supported with an overhang at each end of a=0.5m. A gymnast weighing 50kg is standing a distance c to the right of support A and a distance b to the left of support B. How does the maximum bending moment in the beam depend on the distance c? Express your answer in terms of c assuming units of Nm for the bending moment. Please take the acceleration due to gravity to be 10m/s2.
Mmax: ?
250*c*(2-c)
To find the maximum bending moment in the beam, we need to consider the forces and weight acting on the beam.
First, let's calculate the resultant force of the gymnast's weight. The weight of the gymnast can be calculated as follows:
Weight = mass x acceleration due to gravity
Weight = 50 kg x 10 m/s^2
Weight = 500 N
Now, let's analyze the forces acting on the beam. There are three main forces to consider:
1. The reaction force at support A: This force acts upward to counterbalance the weight of the gymnast. It is equal to the weight of the gymnast, which is 500 N.
2. The reaction force at support B: This force also acts upward to counterbalance the weight of the gymnast. It is also equal to the weight of the gymnast, which is 500 N.
3. The weight of the beam: This force acts downward and is evenly distributed along the length of the beam. It can be calculated as follows:
Weight of the beam = weight per unit length x length of the beam
Since the length of the beam is 3 m and the weight per unit length is the weight of the beam divided by the length of the beam, we get:
Weight of the beam = (Weight of the beam / 3 m) x 3 m
Weight of the beam = Weight of the beam
Now, let's calculate the maximum bending moment (Mmax) at the center of the beam. At the center, the bending moment is maximum. We can determine the value of Mmax by taking moments about the center of the beam.
Taking moments about the center:
Sum of clockwise moments = Sum of counterclockwise moments
Clockwise moments:
Mmax x 0.5 (due to the reaction at support A)
Counterclockwise moments:
500 N x c (due to the gymnast)
Weight of the beam x 1.5 (due to the weight of the beam)
Setting the clockwise moments equal to the counterclockwise moments:
Mmax x 0.5 = 500 N x c + Weight of the beam x 1.5
To express Mmax in terms of c, we can rearrange the equation:
Mmax = 500 N x c + Weight of the beam x 1.5 - Mmax x 0.5
To simplify the equation, we can combine like terms:
1.5 x Weight of the beam = Mmax x (0.5 + 1)
Finally, we get:
Mmax = 1.5 x Weight of the beam / 1.5
Since the weight of the beam is not given in the question, we cannot provide a numerical value for Mmax in terms of c. However, you can substitute the weight of the beam to calculate the maximum bending moment.