Can a crane lift part of itself off the ground? I was looking at some construction work involving a crane and I thought of something...
Two construction workers are driving a crane on the road when suddenly one of the front tire gets a flat.  While they have a spare tire, they don't have a farm jack to raise the front end of the vehicle.  One the workers says I have an idea, I'll raise the boom enough so that it becomes a hypotenuse of a right triangle and I lower the hook on the cable so we can attach it to the bumper.  So the cable and the bumper make a 90 degrees angle with the boom as he hypotenuse.  I will then pull the cable, and, with the pivot at the end of the boom, we should be able to lift the frontend and change the tire.  The second construction workers says, I have a better idea, let's make the cable the hypotenuse with the boom and the pivot at 90 degrees and then lift the front bumper.
Never mind the details such as whether the bumper is strong enough, etc.  Would any of the above plan work to raise the frontend of the crane?  If not, would any other triangle shape work?  I just can't get my head wrapped around this puzzle that I created for myself. Thanks!
 A: If I understand the question correctly, the crane is firmly attached to the chassis of the truck. The bumper is assumed to be firmly attached to the chassis as well. Attaching the hook anywhere on the bumper will not lift the wheels off the ground. 
The hand waving argument: This is like trying to lift yourself off the floor by using your hands to pull up on your own knees. 
The formal argument would be this: The cable provides an upward force on the bumper that is exactly cancelled by the downward force at the tip of the crane. Because of vector addition, no change in angle of the crane can alter this exact cancellation. In practice you could deform the bumper, but never lift the whole truck. 
If I am misunderstanding the question and the crane is attached to the ground and not the chassis, then yes any upward force would lift the front of the vehicle. 
Many cranes have retractable support legs that stabilize the crane laterally. Even if the legs are engaged the crane cannot provide a torque about the rear wheels. However, the support legs often have enough travel and strength to lift the vehicle on their own. In that case, front mounted legs could lift the front of the vehicle. I've seen this done. It may even be standard procedure.
A: I just happened to think of this same scenario the other day.
The answer is no, you cannot use a crane to lift itself (with a few very tricky exceptions). This is because of some really fundamental physics, not just because no one is clever enough to think of a way to do it.
Newton's first law is that an object remains at rest unless some force acts on it. The third law says that forces come in pairs -- when one object exerts a force on another object the first also experiences a force equal and opposite. Together with Newton's second law, this implies that it is impossible to move an object's center of mass without external forces on the object.
By definition, any force the crane exerts on itself is an internal force. Since raising the bumper almost certainly involves moving the crane's center of mass, Newton's laws imply that the crane cannot raise its own tires.

If you are thinking really hard to might have noticed a hole in my argument. It is possible for the crane to raise a part of itself as long as the center of mass doesn't move. For example, if I lay flat on the ground I could (maybe) in principle crunch myself forward and grab my feet and lift them. This is possible as long as all of my movements are just rotations about my center of mass on the ground. However, a crane's center of mass is almost certainly well off the ground so there's no maneuver it could perform to lift itself without moving the center of mass.
A: My intuitive feeling is no, this will not work, because there is no force available to pull the top of the boom backwards and start the tipping process. If such a force was available, then maybe, but I would not offer to try it.
Sorry I can't draw you a picture on this smartphone,  but I think you need to have another force either parallel to the ground, or better still, directed downwards, at the back of the crane.
A: For balance all any object needs is its CG's projection down fall into the area created by three supports which could be wheels or legs.  
If by rotating the crane boom out and lowering it a bit to maximize its overturning moment the CG of the crane will move to the triangle made by the three good wheels and the crane will tilt back level, suspending the flat tire in the air. there is no need to use the cable to lift the bumper any more.  
I have actually done a version of this technic by asking my friends to sit on the opposite corner of my car so that I could replace my flat tire without a jack.
A: Suppose it would work. Then you have a car with a high crane on it standing on its back tires. The only way the car can stand in this (highly unstable) equilibrium is when gravity develops equal torques wrt the line that connects both centers of the back tires (the forces of gravity pulling on one side for the most part on the crane and on the other side on the car). To get the car in that position with the method in your question, then the car must pass positions in which the torque on one side is bigger than on the other side, which makes it already at the start impossible to pull the car up. Either the cable will snap, the crane and car get damaged, or both. 
This situation can only occur though when the car is put in that position (say by attaching it to another crane). But the slightest push upward to or pull down on the car will cause a big crash. 
What could do the trick though (pull up the foreside of the car) is to attach the hook on the end of the crane cable to the Earth's surface. The crane's top part must be aligned with the length of the car. The top part can point forward or backward, but it takes the smallest amount of energy when the crane's top part points backward while maintaining an angle of 90 degrees between the top part and lower part of the crane when pulling the cable in.
