# How do you tell what forces do no work?

The total mass of the children and the toboggan is 66 kg. The force the parent exerts is 58 N (18 degrees above the horizontal). What 3 forces/ components do no work on the toboggan?

I said the vertical component of the force being applied, since the displacement and the force will be perpendicular. I also said gravity and the $x$ component of the force being applied since it will also be perpendicular. I don't know any other things that will make work $0$.

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Think about the way you calculate work as $F dx$. Anything that will make that 0 is a good answer – Michiel Mar 28 '13 at 6:28

Work is the change in energy, $W=\Delta E$. If the sum of the forces on an object (draw the free-body diagram) is non-zero then this may result in a change of either the potential energy, the kinetic energy, or both implying a change in the total energy. If such a net force acts in such a way that neither the kinetic nor potential energy change, such as in the case of a mass sliding on a frictionless surface, held to a center by a string, then no work is done. In this case the force is always perpendicular to the velocity. So while the mass is accelerating, it's kinetic energy isn't changing.

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The three forces components that do no work on the toboggan are:

1. Gravity (since it is acting at right angles to the toboggan motion.)
2. The vertical component of the parents pulling forces (for reason you stated)
3. The reaction force exerted by the ground on the tobogan+children, since it also acts perpendicular to the toboggan's motion.

The x-component of the pulling force does do work.

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I usually take the dot product of the force vector with the instantaneous velocity vector at the point where the force is applied. If $\vec{F}\cdot \vec{v} = 0$ then the force does no work.

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The forces acting on the toboggan are:

1) Vertical force of gravity downward; since there is no vertical velocity, this force does no work.

2) Vertical reaction force from the surface, upward; see 1) above

3) Vertical component of parent force upward; see 1) above

4) Horizontal friction force (snow on toboggan) backward; since this force is acting in the opposite direction to the motion, it does negative work. That is, the toboggan does work on the snow, moving, warming or melting it.

5) Horizontal component of parent force; since this is in the same direction as the motion, this force does work. The work may increase the velocity of the toboggan, or be passed on as work done by the toboggan on the snow, as in 4) above...

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