A bridge of length 22 m and mass 1.5 × 105 kg is supported at each end . A truck of mass 25000 kg is located 17 m from the left support A. Calculate the normal force on the bridge at point B (the right support). The acceleration of gravity is 9.8 m/s 2 .
To calculate the normal force on the bridge at point B, we need to consider the forces acting on the bridge and the truck.
First, let's write down the forces acting on the bridge:
1. Weight of the bridge (W_bridge): This force acts vertically downwards at the center of mass of the bridge. It can be calculated using the formula:
W_bridge = mass_bridge * acceleration_due_to_gravity
Given that the mass of the bridge is 1.5 × 10^5 kg and the acceleration due to gravity is 9.8 m/s^2, we can calculate:
W_bridge = (1.5 × 10^5 kg) * (9.8 m/s^2)
2. Reaction force at point A: Since the bridge is supported at each end, there is a reaction force at each support. At support A, this force acts vertically upwards to balance the weight of the bridge. We will denote this force as N_A.
3. Reaction force at point B: Similarly, at support B, there is a reaction force upwards to balance the weight of the bridge and the truck. We will denote this force as N_B, which is the force we are trying to calculate.
Next, let's consider the forces acting on the truck:
1. Weight of the truck (W_truck): This force acts vertically downwards at the center of mass of the truck. It can be calculated using the formula:
W_truck = mass_truck * acceleration_due_to_gravity
Given that the mass of the truck is 25000 kg and the acceleration due to gravity is 9.8 m/s^2, we can calculate:
W_truck = (25000 kg) * (9.8 m/s^2)
Now, let's consider the forces acting at point B. At this point, there are two downward forces: the weight of the bridge (W_bridge) and the weight of the truck (W_truck). The net force acting downwards at point B can be calculated as the sum of these two forces:
Net downward force at B = W_bridge + W_truck
Finally, the normal force at point B is equal in magnitude but opposite in direction to the net force at point B. Therefore, the normal force at point B (N_B) can be calculated as:
N_B = - (Net downward force at B)
Now, using the calculated values for W_bridge and W_truck, and substituting them into the equation for the net force at B, we can find the normal force at point B.
To calculate the normal force on the bridge at point B, we need to analyze the forces acting on the bridge.
1. First, let's calculate the weight of the bridge:
Weight = mass × acceleration due to gravity
Weight = 1.5 × 10^5 kg × 9.8 m/s^2
Weight = 1.47 × 10^6 N
2. Next, let's calculate the weight of the truck:
Weight of truck = mass × acceleration due to gravity
Weight of truck = 25000 kg × 9.8 m/s^2
Weight of truck = 245000 N
3. Now, let's consider the weight distribution on the bridge. Since the truck is located 17 m from the left support, it creates a clockwise moment about point A. To balance this moment, the bridge needs to exert a counter-clockwise moment.
4. The moment caused by the weight of the bridge is given by:
Moment = weight of bridge × distance from A to B
Moment = 1.47 × 10^6 N × 17 m
Moment = 2.499 × 10^7 Nm
5. To find the normal force at point B, we need to calculate the sum of the vertical forces acting on the bridge at point B.
6. The vertical forces acting on the bridge are the weight of the bridge and the weight of the truck:
Vertical force at B = weight of bridge + weight of truck
Vertical force at B = 1.47 × 10^6 N + 245000 N
Vertical force at B = 1.715 × 10^6 N
Therefore, the normal force on the bridge at point B is 1.715 × 10^6 N.