Question

Can someone explain this?

A 8500kg, 19m long bridge supported by two supports, has a truck on it.

`"Force from front wheels"=1500"N"`
`"Force from rear wheels"=4800"N"`

Distance from support A to front wheels = 4.0m
Distance from front wheels to rear wheels = 2.5m
Distance from rear wheels to support B = 12.5m

Calculate the vertical forces at each of the supports.

The answer is to take all the clockwise moments around A (which I did), which is equal to `828550"Nm"`

However, it then says
Anticlockwise moments about A:
`"F"_a*19"m"=828550"Nm"`
`"F"_a=828550/19=43610"N"`

It then says that `"F"_a+"F"_b=1500"N"+4800"N"+83300"N"=89600"N"`

Why do you take the anticlockwise moments around A to be `"F"_a*19"m"` and that `"F"_a+"F"_b=89600"N"`?

Answer 1

The explanation for both parts of your question is that the bridge is in linear and angular equilibrium.

Explanation:

The bridge/truck system is in mechanical equilibrium. It is standing, and hopefully will continue to stand for years. Mechanical equilibrium means that both linear acceleration and angular acceleration are zero. If those accelerations are to be zero, net force and net torque need to be equal and opposite.

Why do you take the anticlockwise moments around A to be `F_a*19 m`?
[I just now noticed what I hope is a simple typo. The anticlockwise moment around A should be `F_b*19 m`. Then you calculate `F_b`.]

Equilibrium is the reason the anticlockwise moments around A equals `F_b*19 m`. Support B is providing an upward force which causes an anticlockwise torque about A which must be equal and opposite the clockwise torque which you calculated. Otherwise the bridge would be pivoting about A.

Why do you take that `F_a + F_b = 89600 N`?
Equilibrium is the reason that the sum of the upward forces from the 2 supports equals the weight of the bridge+truck. Otherwise, the bridge and truck would be accelerating upward or downward. Not a desirable situation. If you like, you could now calculate `F_a`.

I hope this helps,
Steve