91Ó°ÊÓ

When studying circular motion, energy conservation is a key principle. It implies that the total mechanical energy of the system remains constant as long as no external forces perform work on it, such as air resistance or friction. In the case of the stone being whirled, all changes between kinetic and potential energy happen within the system, without energy loss. This means:

• At the lowest point, the stone has its maximum kinetic energy because its speed is highest.
• As the stone moves up to the horizontal position, some of this kinetic energy is converted into potential energy because it is against gravity.

This conversion does not change the total energy; it merely trades off between kinetic and potential forms. The equation representing energy conservation is:\[ \frac{1}{2}mu^2 = \frac{1}{2}mv^2 + mgL \]This equation allows us to calculate the speed, or velocity, at different points in the circular path using the initial conditions like speed and height.

Vertical circular motion occurs when an object moves along a circular path in a vertical plane. This creates a unique scenario compared to horizontal circular motion due to the presence of gravitational forces acting on the object. In this motion:

• The object experiences a change in gravitational potential energy as it moves up or down.
• Gravity acts as a centripetal force component helping to keep the object in motion.
• The tension in the string varies based on the object's position in the circle.

When the stone is at the lowest point, all force components align to maximize velocity. As it moves upwards, gravity works against the stone, converting kinetic energy into potential energy, slowing the stone down. At the horizontal position, the string is perpendicular to the gravitational force, which crucially influences the tension and speed relationship.

Mechanical energy in physics describes the sum of potential and kinetic energies in a system. For the stone tied to the string, this concept is vital in understanding how energy transforms as it moves:

• Potential energy increases as the stone is raised against gravity.
• Kinetic energy relates directly to the stone's speed; higher speeds mean more kinetic energy.
• In a closed system like our example, potential energy transformations directly affect kinetic energy.

Initially, the stone's mechanical energy is all kinetic at the lowest point with zero potential energy. As it moves to a higher position, some kinetic energy transitions into potential energy, demonstrating mechanical energy's conservation. This balance allows us to compute the velocity at different heights, using potential energy gains to calculate kinetic energy losses.

Understanding the change in velocity is crucial during circular motion because it influences the object's path and speed. In this vertical motion exercise, velocity changes due to the conversion between kinetic and gravitational potential energy:

• When the stone is at the bottom, the velocity is directed vertically with speed \(u\).
• As the stone reaches the horizontal position, this speed reduces to \(\sqrt{u^2 - 2gL}\) due to energy conservation.
• The directions of velocity also change, from vertical to horizontal.

To find the change in velocity's magnitude, consider these perpendicular directions. The formula using the Pythagorean theorem gives the change as:\[ \Delta v = \sqrt{u^2 - (\sqrt{u^2-2gL})^2} = \sqrt{2gL} \]This result shows that the speed change comes from potential energy's conversion to kinetic as the stone rises, resulting in a decrease in the magnitude of velocity from \(u\) to the evaluated horizontal speed.