Moments and Equilibrium

2009-07-01 6:25 am


The left and right wheels of a double-decker bus are 2.0 m apart. Its centre of gravity is
at a height of 2.0m.It is on a road which is slanted by angle a to the left.What is the
maximum value of a if the bus is not to topple?
I have somthing to ask base on this:
1)Why does toppling just avoided when the vertical from the centre of gravity X pass
through the tyre on the lower side?

2)Although the moment due to weight about the lower tyre is eliminated,the normal
reaction on the upper tyre will also cause an antclockwise moment about the lower tyre.
Then why toppling is avoided?

回答 (1)

2009-07-01 7:49 pm
✔ 最佳答案
Consider the very moment the bus just starts to topple. At that point in time, let us take moment about the lower tyre. What forces we have to consider?
1. The weight of the bus.
2. The normal reaction at the upper tyre
3. The friction at the upper tyre

Now since we say the bus just starts to topple, that means the upper tyre is not pressed against the road. What does it means? It mean the normal reaction at the upper tyre is zero, so is the friction. If the line of force from the centre of gravity just passes through the lower tyre, then the moment due to weight is also zero. Total net moment =0 means the bus will not topple.

When the slant angle is a bit bigger, you will see a net anti-clockwise moment about the lower tyre due to entirely the weight of the bus causing the bus to topple.

When the slant angle is a bit smaller, the normal reaction and friction at the upper tyre is non-zero, the moment caused by these two forces will balance the moment caused by the weight of the bus. The bus will not topple.

Let us make a proposition that the bus will indeed topple at the smaller angle. As explained earlier, the normal reaction and the friction will vanish, leaving a net clockwise moment about the lower tyre due to the weight of the bus. This is a conctradiction to the proposition.

Therefore the answer to your second question, the normal reaction at the upper tyre is zero at the critical slant angle.


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