Conservation of Energy for AP Physics B & C

Practice problems for these concepts can be found at: Energy Conservation Practice Problems for AP Physics B & C

Conservation of Energy: Problem-Solving Approach

Solving energy-conservation problems is relatively simple, as long as you approach them methodically. The general approach is this: write out all the terms for the initial energy of the system, and set the sum of those terms equal to the sum of all the terms for the final energy of the system. Let's practice.

If we were to approach this problem using kinematics equations (which we could), it would take about a page of work to solve. Instead, observe how quickly it can be solved using conservation of energy.

We will define our zero of potential to be the height of the box after it has slid the 5 m down the plane. By defining it this way, the PE term on the right side of the equation will cancel out. Furthermore, because the box starts from rest, its initial KE also equals zero.

The initial height can be found using trigonometry: h_i = (5m) (sin 30°) = 2.5 m.

In general, the principle of energy conservation can be stated mathematically like this:

The term W in this equation stands for work done on an object. For example, if there had been friction between the box and the plane in the previous example, the work done by friction would be the W term. When it comes to the AP exam, you will include this Wterm only when there is friction involved. When friction is involved, W = F_fd, where F_f is the force of friction on the object, and d is the distance the object travels.

Let's say that there was friction between the box and the inclined plane.

We start by writing the general equation for energy conservation:

W equals F_fd, where F_f is the force of friction, and d is 5 m.⁴

The value for W is negative because friction acts opposite displacement. You may want to draw a free-body diagram to understand how we derived this value for F_N.

Now, plugging in values we have

We rearrange some terms and cancel out m from each side to get

v_f = 5.7 m/s

This answer makes sense—friction on the plane reduces the box's speed at the bottom.

Springs

Gravitational potential energy isn't the only kind of PE around. Another frequently encountered form is spring potential energy.

The force exerted by a spring is directly proportional to the amount that the spring is compressed. That is,

In this equation, k is a constant (called the spring constant), and x is the distance that the spring has been compressed or extended from its equilibrium state.⁵

When a spring is either compressed or extended, it stores potential energy. The amount of energy stored is given as follows.

Similarly, the work done by a spring is given by W_spring = ½kx². Here's an example problem.

It's important to recognize that we CANNOT use kinematics to solve this problem! Because the force of a spring changes as it stretches, the block's acceleration is not constant. When acceleration isn't constant, try using energy conservation.

We begin by writing our statement for conservation of energy.

Now we fill in values for each term. PE here is just in the form of spring potential energy, and there's no friction, so we can ignore the W term. Be sure to plug in all values in meters!

Plugging in values for k and m, we have

v_f = 0.07 m/s, that is, 7 cm/s.

 

Power

Whether you walk up a mountain or whether a car drives you up the mountain, the same amount of work has to be done on you. (You weigh a certain number of newtons, and you have to be lifted up the same distance either way!) But clearly there's something different about walking up over the course of several hours and driving up over several minutes. That difference is power.

Power is, thus, measured in units of joules/second, also known as watts. A car engine puts out hundreds of horsepower, equivalent to maybe 100 kilowatts; whereas, you'd work hard just to put out a power of a few hundred watts.

Practice problems for these concepts can be found at: Energy Conservation Practice Problems for AP Physics B & C