A box of mass m is placed on a plane inclined at a 30° angle above the horizontal. The coefficient of
friction between the box and the plane is 0.20. The box is released from rest and allowed to slide 5.0 m
down the plane. What is its final velocity?We start by writing the general equation for energy conservation:
Ki + Ui + W = Ef + UfW equals F (^) f d , where F (^) f is the force of friction, and d is 5 m.^2
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. The negative sign is simply a reminder that the force
of a spring always acts opposite to displacement—an extended spring pulls back toward the equilibrium
position, while a compressed spring pushes toward the equilibrium position. We call this type of force a
restoring force, and we discuss it more in Chapter 17 on simple harmonic motion. However, we can
ignore this sign unless we are doing calculus.
When a spring is either compressed or extended, it stores potential energy. The amount of energy