A heavy metal beam AB of mass 25Kg is supported at its ends.The beam carries a mass of 150Kg at a distance of 0.75m from end A.If the beam is 2m long, determine the thrust A and B.
To determine the thrust at points A and B, we need to calculate the reactions at those points.
Let's start by calculating the total weight of the beam and the load it carries.
The weight of the beam itself is given by:
Weight_beam = mass_beam * acceleration_due_to_gravity
Weight_beam = 25 kg * 9.8 m/s^2
Weight_beam = 245 N
The weight of the load is given by:
Weight_load = mass_load * acceleration_due_to_gravity
Weight_load = 150 kg * 9.8 m/s^2
Weight_load = 1470 N
Next, we can calculate the total weight acting on the beam:
Total_weight = Weight_beam + Weight_load
Total_weight = 245 N + 1470 N
Total_weight = 1715 N
Since the beam is in equilibrium, the sum of the vertical forces acting on it must be zero.
Therefore, the reaction at point A (thrust A) can be determined by taking moments about point B and setting them equal to zero.
Taking moments about point B:
(Reaction at B * distance from B to load) - (Total weight * distance from B to center of gravity of beam) = 0
(Reaction at B * 0.75 m) - (1715 N * 1 m) = 0
0.75 * Reaction at B - 1715 = 0
0.75 * Reaction at B = 1715
Reaction at B = 1715 / 0.75
Reaction at B = 2286.67 N
Since the sum of the reactions is equal to the total weight, we can calculate the reaction at point A (thrust A) as follows:
Reaction at A = Total weight - Reaction at B
Reaction at A = 1715 N - 2286.67 N
Reaction at A = -571.67 N
Note that the negative sign indicates that the reaction at point A is upward.
Therefore, the thrust at point A is -571.67 N, and the thrust at point B is 2286.67 N.
To determine the thrust at points A and B, we need to analyze the forces acting on the beam and apply the principle of equilibrium. The thrust at each point refers to the upward force exerted on the beam to counterbalance the weight of the beam and the load.
1. Start by drawing a diagram of the beam, labeling the given values, and indicating the forces acting on it. In this case, we have:
- Mass of beam, m1 = 25 kg
- Mass of load, m2 = 150 kg
- Distance of load from end A, d = 0.75 m
- Length of the beam, L = 2 m
A B
|--------|
<----L---->
|--------|
|---m2---|
- Forces: weight of beam (W1), weight of load (W2), thrust at point A (A), thrust at point B (B)
2. Calculate the weights of the beam and load using the formula W = m * g, where g is the acceleration due to gravity (approximately 9.8 m/s^2):
- W1 = m1 * g = 25 kg * 9.8 m/s^2
- W2 = m2 * g = 150 kg * 9.8 m/s^2
3. Since the beam is in equilibrium, the sum of vertical forces and the sum of the moments about any point must equal zero.
Sum of vertical forces: A + B - W1 - W2 = 0
4. To calculate the moment, we need to choose a point about which to take moments. Taking moments about point A simplifies the calculation since B will have no contribution to the moment equation.
Sum of moments about point A: (B * L) - (W1 * (L/2)) - (W2 * (L - d)) = 0
Breaking the equation down:
- B * L = moment contributed by thrust at point B
- W1 * (L/2) = moment contributed by weight of beam (W1) at its center
- W2 * (L - d) = moment contributed by weight of load (W2) at its distance from A
5. Now, we have two equations (from steps 3 and 4) with two unknowns (A and B). We can solve these equations simultaneously to find the values of A and B.
From the vertical force equation:
A + B - W1 - W2 = 0 (equation 1)
From the moment equation:
(B * L) - (W1 * (L/2)) - (W2 * (L - d)) = 0 (equation 2)
Substitute the known values into the equations and solve for A and B.
Once you plug in the values and solve the system of equations, you can find the values of thrust at points A and B, which will provide the upward forces needed to balance the weight of the beam and the load.
A------m--M-----------B
0.75
M=25kg
m=150kg
Distance AM=2.0/2=1.0 m
Take moments about A:
(counter-clockwise positive)
B*2-150*0.75-25*1.0=0
B=(150*0.75+25)/2=137.5/2=68.75 kg
The reaction (thrust) at A can be found by summing vertical forces and equating to zero, or similarly taking moments about B.