91Ó°ÊÓ

Kinetic energy is the energy that an object has due to its motion. When a ball is at rest, like the 10 kg ball at a point on the x-axis, its kinetic energy is zero. However, as it starts to move, this energy increases. Kinetic energy can be calculated using the formula:

• \( KE = \frac{1}{2}mv^2 \)

where \( m \) is the mass of the object and \( v \) is its velocity.
This formula shows that kinetic energy is directly proportional to the mass and the square of the velocity. In our exercise, energy conservation plays a role. As the ball moves from a point of zero potential energy at infinite distance to the point (0.30 m, 0), it gains kinetic energy equal to the potential energy lost. This change is due to gravitational forces acting on it.
So the greater the velocity or mass, the higher the kinetic energy acquired, as depicted by the calculation at the final point, resulting in the ball's kinetic energy being significant towards the end of its path.

Gravitational force is a force of attraction that exists between any two masses. This force is what pulls objects towards one another. The strength of the gravitational force between two objects depends on their masses and the distance between them.
The gravitational force can be calculated with the formula:

• \( F = \frac{Gm_1m_2}{r^2} \)

where \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses, and \( r \) is the distance between them.
In the given exercise, the 10 kg ball is under the influence of gravitational forces from two spheres, each weighing 20 kg, which are located symmetrically about the x-axis. These forces act in opposite directions along the y-axis. Therefore, their net result in the y-direction cancels out. However, they contribute to the ball's kinetic energy through potential energy changes.

Energy conservation is a principle stating that energy cannot be created or destroyed; it can only be transformed from one form to another. In physics, this means that the total energy of an isolated system remains constant over time.
For our exercise, when the ball starts moving towards the point (0.30 m, 0), energy conservation dictates that the potential energy lost due to the gravitational forces from the spheres is transformed into kinetic energy. Initially, at an infinite distance, the potential energy is zero. As it approaches the two spheres, it converts this potential energy into kinetic energy while maintaining total energy.
This principle helps us understand that any potential energy reduction in the system must equivalently increase the kinetic energy of the ball, ensuring the total energy remains constant throughout its motion.

Newton's Law of Universal Gravitation explains how every point mass attracts every other point mass in the universe with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The mathematical expression for this is:

• \( F = \frac{Gm_1m_2}{r^2} \)

where \( F \) is the gravitational force, \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses, and \( r \) is the distance between the centers of their masses.
This universal law applies to all objects with mass. In our exercise, it helps us calculate the gravitational forces exerted by the two 20 kg spheres on the 10 kg ball. Despite their symmetrical placement, they exert equal and opposite forces on the ball along the y-axis leading to a net zero force. This understanding is crucial in predicting motion under gravitational influence, allowing us to determine resultant forces and energy transformations that take place.