Help! How would you solve this physics problem? I don't even know how to begin.

Question

A person wants to push a lamp of mass m across a truly hideous carpet with coefficient of friction u. let the radius of the lamp's base be r, its mass m, and the height . of the center of mass h. In terms of h, how high up can the person push the lamp and make it move without tipping? Start off by guessing what the answer might be, on a combination of physical and dimensional grounds.

Answer 1

Start by drawing a picture.

There are 3 forces acting on the lamp:

gravity

friction (depends on gravity force)

applied force when you push it

The gravity force is the mass times the acceleration of gravity: F(g) = m * 9.8 (using MKS units)

The friction force is F(g) * u = F(f)

If you push at the base of the lamp, then your applied force F(a) is offset by F(f). If F(a) is larger than F(f), the lamp will accelerate.

If you push at a point above the base, the same rules apply with respect to friction. There is also now a torque (AKA moment), which acts to try and tip the lamp over.

Let's assume that F(a) and F(f) are the same. This means that the lamp is either not sliding or is sliding at a constant speed. The lateral forces are in balance, but there is still a torque (T(a)) with is F(a) * d, where d is the height of the applied force. There is an opposite torque determined by the weight of the lamp and the lateral distance from the center of mass to the edge of the base. The lateral distance is simply the radius of the base (r), so the opposing torque is simply F(g) * r.

Note that the height of the center of mass is not relevant until the lamp starts to tip. At this point, the lever arm starts to shorten and there is now less torque associated with the weight of the lamp. At the same time the lever arm to the applied force starts to get longer, meaning that more force is now required to continue tipping.

So--lay all this out on a drawing. Then do some experiments with an actual object to see how things behave and how the various forces interact

Additional comments 11/17:

Think about what actually happens when you start pushing the lamp at a point above the base.

With no friction, it will simply start sliding---and it will not matter how high your applied force is.

WITH friction, the applied force results in a torque (moment), that tends to tip the lamp.

First--assume that the friction is high enough that the lamp does not slide. As you increase the applied force, the torque causes the lamp to tip. Initially, the required torque is the weight of the lamp times the horizontal distance from the center of mass to the fulcrum, which is the edge of the base.

Once the lamp starts to tip, then the lever arm changes---for both the weight and for the applied force. Initially, more force is required to increase the tipping angle, but eventually less force is required. When the center of mass is directly above the fulcrum, no force is required--- a slight additional push will then cause the lamp to fall over. (Prove this to yourself by pushing something like a spray can when it is resting on a high-friction surface so it cannot slide)

Next, consider the case where the applied force causes the lamp to start sliding. Now, it is the "net force" that will cause the lamp to move at a constant speed, accelerate, and/or tip. To analyze this, we will need to make assumptions about how the coefficient of friction changes as the lamp tips. In all cases, the "net force" is simply the difference between the applied force and the friction force. The net force is what causes the lamp to accelerate, tip, or both. And the tipping is determined by the torque and thus depend on the angle.