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: ?
To find the maximum bending moment (Mmax) in the beam, we need to consider the moments caused by the gymnast's weight on both sides of the beam.
First, let's calculate the magnitude of the moment caused by the gymnast's weight on the right side of the beam:
Moment caused by the gymnast's weight on the right side = (Weight of the gymnast) × (Distance between the center of gravity and support A)
Given:
- Weight of the gymnast = 50 kg
- Distance c to the right of support A
The distance between the center of gravity and support A can be calculated as follows:
- Total length of the beam = 3 m
- Overhang on each end = 0.5 m
- Distance between support A and the center of the beam = (Total length of the beam - Overhang on each end) / 2 = (3 - 2 × 0.5) / 2 = 1 m
So, the distance between the center of gravity and support A = 1 m + c
Now, we can calculate the moment caused by the gymnast's weight on the right side of the beam:
Moment caused by the gymnast's weight on the right side = (Weight of the gymnast) × (Distance between the center of gravity and support A)
= 50 kg × 10 m/s^2 × (1 m + c)
Next, let's calculate the magnitude of the moment caused by the gymnast's weight on the left side of the beam:
Moment caused by the gymnast's weight on the left side = (Weight of the gymnast) × (Distance between the center of gravity and support B)
The distance between the center of gravity and support B can be calculated as follows:
- Total length of the beam = 3 m
- Overhang on each end = 0.5 m
- Distance between support B and the center of the beam = (Total length of the beam - Overhang on each end) / 2 = (3 - 2 × 0.5) / 2 = 1 m
So, the distance between the center of gravity and support B = 1 m - b
Now, we can calculate the moment caused by the gymnast's weight on the left side of the beam:
Moment caused by the gymnast's weight on the left side = (Weight of the gymnast) × (Distance between the center of gravity and support B)
= 50 kg × 10 m/s^2 × (1 m - b)
The maximum bending moment (Mmax) in the beam is the sum of these two moments:
Mmax = Moment caused by the gymnast's weight on the right side + Moment caused by the gymnast's weight on the left side
= 50 kg × 10 m/s^2 × (1 m + c) + 50 kg × 10 m/s^2 × (1 m - b)
Therefore, the maximum bending moment in the beam depends on the distance c as:
Mmax = 500 N × (1 + c) + 500 N × (1 - b)
= 500 N × (2 + c - b)
Note: The unit for the bending moment is Nm (Newton-meter).
To determine how the maximum bending moment (Mmax) in the beam depends on the distance c, we need to consider the weight of the gymnast and the equilibrium condition of the beam.
First, let's calculate the total weight of the gymnast. Given that the gymnast weighs 50 kg and the acceleration due to gravity is 10 m/s^2, we can find the weight (W) using the formula:
W = m * g
where
W is the weight,
m is the mass, and
g is the acceleration due to gravity.
Substituting the given values, we have:
W = 50 kg * 10 m/s^2
W = 500 N
Next, let's visualize the equilibrium condition of the beam. At the maximum bending moment, the forces acting upwards and downwards on the beam must balance. This occurs when the weight of the gymnast is balanced by the reaction forces exerted by the overhangs.
Considering the distances involved, we can split the weight W into two forces, one acting upwards at support A and one acting downwards at support B. Let's denote these forces as R1 and R2, respectively.
Now, at support A, we have the reaction force R1 acting upwards. The distance between the gymnast and support A is c. At support B, we have the reaction force R2 acting downwards. The distance between the gymnast and support B is b.
To find the values of R1 and R2, we can use the principle of moments. The total moment around support A must be equal to zero since the beam is in equilibrium.
Mathematically, the sum of clockwise moments should be equal to the sum of counterclockwise moments:
(R1 * a) - (W * c) = 0
Simplifying the equation:
R1 * a = W * c
Similarly, using the same principle of moments at support B:
(R2 * a) - (W * b) = 0
Simplifying this equation:
R2 * a = W * b
Now, substituting the value of W:
R1 * a = 500 N * c
R2 * a = 500 N * b
We can solve these two equations to find the values of R1 and R2. However, since we are interested in the maximum bending moment (Mmax) in the beam, we need to consider the scenario where either R1 or R2 is at its maximum.
The maximum bending moment occurs when either R1 or R2 is at its maximum value, resulting in the beam being loaded at one support only. In this case, the equation for the bending moment (M) can be formulated as:
M = (R1 or R2) * (a + (c or b))
where (R1 or R2) is the maximum value of either R1 or R2, and (c or b) is the distance of the gymnast from either support A or B, whichever corresponds to the maximum value.
Therefore, the maximum bending moment (Mmax) in the beam depends on the distance c as follows:
Mmax = R1 * (a + c)
Note that for different values of c, the maximum bending moment may occur at either support A or support B. However, since you specifically asked for the dependence on c, I have assumed that the maximum bending moment occurs at support A for simplicity. If you want to consider the case where the maximum bending moment occurs at support B, simply replace c with b in the above equation.