# Work-energy principle: What if there is no change in KE, but change in PE?

So I've been taught that the work-energy principle says that the total work done on an object is always equal to the change in kinetic energy of an object.

Now let's say that I have a car moving up an incline at constant speed. The kinetic energy does not change from point A to point B, but the potential energy does, right? Yet there is friction on the slope, so there was some work done against friction, and the engine is providing a driving force, so there is work done by the driving force.

What am I missing here?

The Work-KE theorem says the net work on a system is equal to its change in KE. But the net work includes work done by conservative and nonconservative forces, and the work done by conservative forces is, by definition, the negative of a change in PE:

$$W_{net} = \Delta KE = W_C + W_{NC}$$

$$=-\Delta PE + W_{NC}$$,

so $$W_{NC} = \Delta (KE + PE) = \Delta E_{mech}$$.

(Hence the name: work done by nonconservative forces does not conserve the mechanical energy of the system.)

In your example, the forces doing work on the car as it climbs a hill at constant speed are its weight (doing negative work) and friction from the road (doing positive work). The weight is a conservative force, so work it does corresponds to the change in the car's PE (i.e. $$W_{mg} = -mg\Delta h$$, so $$∆PE = mg∆h$$). Because the car is moving at constant speed you know the friction force balances out the component of the weight down the incline, so the work done by these forces have equal magnitude. It is a nonconservative force, and $$W_{fric} = +mg\Delta h$$. These add to zero net work, so zero change in KE, but $$W_{NC} = mg\Delta h = \Delta PE = \Delta E_{mech}$$ (because $$\Delta KE = 0$$).

(Note that the important thing is not the work done against friction, but the work done by friction - this is what's pushing the car up the slope, after all. Also, the engine does no work on the car, since it's part of the car and not an external agent, but it does provide the torque on the wheels that generates the friction force.)

Work is done by driving force but work energy theorem is all about work on the object and in your example it is your car on which work is being done which is equal to zero because kinetic energy doesn't change or in other words total force acting on the object is zero all the time because you consider velocity to be constant.

Now let's say that I have a car moving up an incline at constant speed. The kinetic energy does not change from point A to point B, but the potential energy does, right?

Correct. But since the car is moving at constant velocity up the incline, that means, according to the work energy theorem, the net force acting on the car parallel to the incline is zero, as discussed below. The work-energy theorem which states the change in kinetic energy of an object equals the net work done on the object. But that does not mean the potential energy doesn't change, as discussed in the last paragraph below.

Yet there is friction on the slope, so there was some work done against friction, and the engine is providing a driving force, so there is work done by the driving force.

Work is not done against friction. Work is done against gravity acting down the incline. Work is done by static friction between the road and drive wheels acting up the incline against the downward force of gravity on the car, enabling the car to move up the incline without slipping as long as the maximum static friction force is not exceeded. See the diagram below.

The car drive train generates torque on the drive wheels (not shown in diagram) creating a force acting down the incline. Per Newton's third law, static friction from the road exerts and equal and opposite force acting on the car up the incline, as shown in the diagram. Opposing this force is the component of the force of gravity acting on the car down the incline in the opposite direction of the static friction force. When the downward force of gravity equals the upward static friction force, the net force acting parallel to the incline is zero and the velocity is constant per the work energy theorem.

The net work done is zero because gravity does negative work equal to the positive work done by the static friction force. Positive work transfers energy to the car. The work done by gravity is negative work because its force is in the opposite direction to the movement of movement of the car. Negative work means gravity takes the energy given the car by the car drive drain away and stores it as gravitation potential energy of the car-earth system. The net result is the work done on the car to move it up the incline is stored as gravitational potential energy without a change in kinetic energy (change in velocity)

Hope this helps.

• The static friction force does not, in general, equal $\mu_s M g$.
– pwf
Commented Oct 16, 2020 at 17:41
• Right. I meant it as the maximum static friction force. Thx, I will clarify. Commented Oct 16, 2020 at 17:44