Non-Conservative Forces

As you read, pay attention to Figure 7.15 for a comparison of conservative and non-conservative forces. In Figure 7.15 (a), a rock is being "bounced" on an ideal spring with no friction. The mechanical energy does not change, and the rock will continue bouncing indefinitely. In Figure 7.15 (b), the rock is thrown and lands on the ground. When it hits the ground, its kinetic energy is converted to thermal energy and sound. The rock can not "bounce" back up because its mechanical energy is not conserved.

Gravity is a good example of a conservative force we use a lot in physics. Gravitational force is a conservative force because the work gravity does on an object does not depend on the path the object takes. Consequently, gravity is a good candidate to add into the work-energy theorem, where work is only done by gravity: $W=Fd=mad$

Since the acceleration due to gravity is simply $g$ and the direction of motion due to gravity is in the y-axis, we can further build the equation that represents work due to gravity: $W=mg(\Delta y)=\Delta(mgy)$

Previously, we have discovered that work is also equal to the change in kinetic energy (see Section 7.2). So, we can now combine our equation for work due to gravity and our equation for work with respect to the change in kinetic energy: $\Delta(mgy)=\Delta(\frac{1}{2})mv^{2}$ . The $mgy$ in the equation is called the gravitational potential energy. We define potential energy as stored energy due to a system's position: $PE=mgy$ .

Making Connections: Take-Home Investigation—Determining Friction from the Stopping Distance

This experiment involves the conversion of gravitational potential energy into thermal energy. Use the ruler, book, and marble from Take-Home Investigation – Converting Potential to Kinetic Energy. In addition, you will need a foam cup with a small hole in the side, as shown in Figure 7.19. From the 10-cm position on the ruler, let the marble roll into the cup positioned at the bottom of the ruler. Measure the distance $d$ the cup moves before stopping.

What forces caused it to stop? What happened to the kinetic energy of the marble at the bottom of the ruler? Next, place the marble at the 20-cm and the 30-cm positions and again measure the distance the cup moves after the marble enters it. Plot the distance the cup moves versus the initial marble position on the ruler. Is this relationship linear?

With some simple assumptions, you can use these data to find the coefficient of kinetic friction $\mu_{\mathrm{k}}$ of the cup on the table. The force of friction $f$ on the cup is $\mu_{\mathrm{k}} N$ , where the normal force $N$ is just the weight of the cup plus the marble. The normal force and force of gravity do no work because they are perpendicular to the displacement of the cup, which moves horizontally. The work done by friction is $fd$ . You will need the mass of the marble as well to calculate its initial kinetic energy.

It is interesting to do the above experiment also with a steel marble (or ball bearing). Releasing it from the same positions on the ruler as you did with the glass marble, is the velocity of this steel marble the same as the velocity of the marble at the bottom of the ruler? Is the distance the cup moves proportional to the mass of the steel and glass marbles?

A marble is rolling down a makeshift ramp consisting of a small wooden ruler propped up on one end at about a thirty degree angle. At the bottom of the ramp is a foam drinking cup standing upside-down on its lip. A hole is cut out on one side of the cup so that the marble will roll through the hole when it reaches the bottom of the ramp.

Figure 7.19 Rolling a marble down a ruler into a foam cup.