A uniform plank AB is 10m long and has mass of 14km. The plank rest on two support at A and B. A lead of mass 8kg is placed on the plank at a point c 4meter away from A. Calculate the reactions at the support at A and B on the plank
I'm sure this question is related with moment of force about the supports but still even I'm confused how to solve..need help with this..!!
To calculate the reactions at the supports A and B, we need to consider the forces acting on the plank.
1. Let's consider the forces acting vertically:
- The weight of the plank acts downwards and can be calculated as W_plank = mass_plank * gravitational acceleration = 14 kg * 9.8 m/s^2 = 137.2 N.
- The weight of the lead also acts downwards and can be calculated as W_lead = mass_lead * gravitational acceleration = 8 kg * 9.8 m/s^2 = 78.4 N.
- The reaction at support B acts upwards, balancing the weight of the plank and the lead.
2. Next, let's consider the forces acting horizontally:
- There are no horizontal forces acting, so the reaction at support B is the only force acting in the horizontal direction.
Now, let's calculate the reactions at supports A and B:
1. The vertical forces must balance: W_plank + W_lead = A_vertical + B_vertical.
A_vertical + B_vertical = 137.2 N + 78.4 N = 215.6 N.
2. The horizontal forces must balance: B_horizontal = 0 N.
3. Since the weight of the plank is evenly distributed, the reactions at supports A and B should be equal (symmetry). So, A_vertical = B_vertical = 215.6 N / 2 = 107.8 N.
Therefore, the reaction at support A is 107.8 N and the reaction at support B is 107.8 N.
To calculate the reactions at the supports A and B on the plank, we need to consider both the forces and moments acting on the plank.
First, let's consider the forces acting on the plank. There are three forces to consider: the weight of the plank (W_plank), the weight of the lead (W_lead), and the reactions at the supports (R_A and R_B). The sum of vertical forces in equilibrium gives us:
R_A + R_B - W_plank - W_lead = 0
Since the weight of an object is given by the product of its mass and the acceleration due to gravity (9.8 m/s^2), we can determine the values of W_plank and W_lead:
W_plank = mass_plank * g = 14,000 kg * 9.8 m/s^2 = 137,200 N
W_lead = mass_lead * g = 8 kg * 9.8 m/s^2 = 78.4 N
Now, let's consider the moments acting on the plank. The moment of a force is the product of the force magnitude and the perpendicular distance from the point of rotation. In this case, the point of rotation is at the support A.
Taking moments about point A, we have:
Total clockwise moments = Total counterclockwise moments
(R_B * 10 m) - (W_plank * (10 m / 2)) - (W_lead * 4 m) = 0
Simplifying this equation, we get:
R_B = (W_plank * (10 m / 2) + W_lead * 4 m) / 10 m
= (137,200 N * 5 m + 78.4 N * 4 m) / 10 m
= 686,000 N * m / 10 m + 313.6 N * m / 10 m
= 68,600 N + 31.36 N
= 68,631.36 N
Substituting the value of R_B into the first equation, we can find the value of R_A:
R_A + 68,631.36 N - 137,200 N - 78.4 N = 0
R_A = 137,200 N + 78.4 N - 68,631.36 N
= 68,646 N
Therefore, the reaction at support A is 68,646 N and the reaction at support B is 68,631.36 N.