Maybe I should just turn this blog into the post-Car-Talk discussion forum. In this week's episode of Car Talk, Tom and Ray talk to a lady that would like to turn her Subaru into a snow plow for her driveway. They claim that her idea will probably work, but she won't be able to plow more than like 2 inches of snow at a time. Their reason: her car is not heavy enough. It will just slip when trying to push the snow.
But wait. If you take two blocks (of the same material but different mass) and slide them on a surface, won't they stop at the same distance? First, let me assume the usual model for the frictional force (kinetic friction in this case) that says:
This says that the magnitude of the frictional force is equal to the product of some frictional coefficient (μk) and teh normal force between the block and the surface. In this model, the frictional coefficient depends just on the two types of surfaces rubbing together.
Suppose I now push two different sized blocks so that they start with the same speed. Here is a force diagram for those blocks.
I didn't put it on there, but these blocks are moving to the right. What is the acceleration of block A in the x-direction? First, I need to find the normal force. In the y-direction, the block does not change it's motion. This means that the net force in the y-direction must be zero so that:
So, in the x-direction of motion I have:
The acceleration does not depend on the mass in this case. Block A and B would have the same acceleration.
Now imagine that block A is a snow plow that is pushing some snow of mass msnow. If it is moving at a constant speed and not slipping, then I can draw the following force diagram. Oh, quick note: in this case the frictional force is going to be static friction. I will assume the snow plow is pushing as much snow as it can without slipping. Also, I will assume that the coefficient of kinetic friction between the snow and the road is the same as the static friction coefficient between the tires and the road (just to make things simpler).
Using ideas from above, if the snow is moving at a constant speed then the frictional force on the snow would be equal to the force the snow plow pushes on the snow. This would also be the force the snow pushes back on the plow (forces come in pairs you know). This gives a force of snow pushing on the plow as:
If the net force on the plow is zero (in both y- and x-directions), then in the x-direction:
A couple of notes:
- Yes. I know. The frictional coefficients would not be the same. That is not the point
- The amount of snow the plow can push depends on (even if they are not equal) the mass of the plow)
- If there is no snow, the plow doesn't even need a frictional force to move at a constant speed.
- This solution is just for flat surfaces.
And there you go. More massive snow plows can push more snow.
