91Ó°ÊÓ

Potential energy is the stored energy in an object due to its position or height. In this exercise, both the rat and the mouse start at a height when the rod is horizontal. This gives them potential energy because they are elevated above a reference point, in this case, the lowest point in their swing.
To calculate potential energy, we use the formula:

• \(PE = mgh\)

Where \(m\) is the mass of the object (in kilograms), \(g\) is the gravitational acceleration (approximately \(9.8 \, \text{m/s}^2\) on Earth), and \(h\) is the height above the reference point (measured in meters).
In this scenario:

• The rat has a mass of 0.500 kg and starts at a height of 0.4 m.
• The mouse has a mass of 0.200 kg and also starts at a height of 0.4 m.

By substituting these values into the potential energy formula, we get the initial potential energy for each animal, which can be summed to give the total initial potential energy of 2.744 Joules.

Kinetic energy is the energy of motion. As the rod swings down to a vertical position, the rat and mouse gain speed and thus kinetic energy. The initial potential energy they had is converted entirely into kinetic energy at the bottom of the swing. This happens because energy is conserved, meaning total energy remains constant if there are no external forces acting on the system besides gravity.
The formula to determine kinetic energy is:

• \(KE = \frac{1}{2}mv^2\)

Where \(m\) is the mass of the object and \(v\) is its velocity.
For both animals at the ends of the rod:

• Their combined mass is 0.700 kg.
• The potential energy at the top converts to kinetic energy at the bottom.

Thus, by setting the total kinetic energy equal to the initial potential energy, we solve for their speed in the vertical position, finding that each has a speed of 2.8 m/s.

Rotational motion occurs when an object spins around an axis. In this case, the wooden rod serves as the axis of rotation, with the rat and mouse hanging at each end. Understanding rotational motion helps us determine how the objects behave as they swing from horizontal to vertical.
A key point in rotational motion is that linear speed at the ends of a rotating object depends on the object's angular speed and radius from the axis. However, for our problem, both animals travel the same distance from the pivot to the ends of the rod—0.4 m—and share the same speed when the rod is vertical.
Conservation of energy allows us to figure out this motion without directly dealing with rotation formulas. Still, it's fascinating that rotational motion causes different energy types (potential to kinetic) to interchange, illustrating principles like:

• Moment of inertia, which might increase if more mass is shifted further from the pivot, affecting energy distribution.
• Every point on the rod travels in a circular path determined by the pivot at the center.

Even though side-to-side rotation isn't explicitly necessary in this problem, it's a crucial foundation for more complex rotational dynamics.