A uniform, solid cube of mass \(m\) and side \(s\) is in stable equilibrium when sitting on a level tabletop. How much energy is required to bring it to an unstable equilibrium where it's resting on its corner?

The center of mass is a crucial concept in physics representing the average location of all the mass in a body. For a uniform, solid object like a cube, the center of mass is at the geometric center when it is oriented symmetrically. When sitting flat on a tabletop, a cube’s center of mass is exactly halfway between the top and bottom faces, at a height of \( \frac{s}{2} \) above the table.

In the case of the cube resting on a corner, however, the center of mass shifts. Here, it aligns with the body diagonal of the cube, which means the new height above the table is \( \frac{s\sqrt{3}}{2} \). This diagonal is the longest straight line that can be drawn through the cube, and finding the center of mass along this line involves a more complex geometric consideration, reflecting the cube’s symmetry along all three spatial dimensions.

Understanding where the center of mass lies is essential for analyzing the cube’s equilibrium and calculating the energy required to alter its position.

Gravitational potential energy is the energy that an object possesses because of its position in a gravitational field. Essentially, the higher an object is lifted against gravity, the more potential energy it gains. The formula to compute this energy is \( W = mgh \), where \( m \) is the mass of the object, \( g \) is the acceleration due to gravity (approximately \( 9.81 \text{m/s}^2 \) on Earth), and \( h \) is the height above the reference point.

In the case of moving the cube to rest on its corner, the increase in height \( h \) from the original stable equilibrium position to the new unstable equilibrium position represents the increase in gravitational potential energy. Precisely calculating this increase requires determining the change in height of the cube’s center of mass as detailed in the previous section.

An object is in stable equilibrium if, when displaced slightly, it tends to return to its original position. Picture a valley – if you move a ball within the valley, it will roll back down to the bottom. The cube sitting flat on its side is in stable equilibrium; any disturbance would be corrected by gravity, pulling it back to rest.

In contrast, unstable equilibrium occurs when a slight disturbance causes the object to move away from its equilibrium position. Imagine balancing a ball on a hill's peak – a slight push and the ball will roll away. The cube balanced on its corner represents an unstable equilibrium because any small perturbation would topple it. The energy required to transition the cube from the stable to the unstable equilibrium is tied to the gravitational potential energy – as the cube’s height is elevated, the energy to hold it in place increases, creating an environment where the slightest nudge can lead to a shift in position.