91Ó°ÊÓ

Before the collision occurs, we must consider the principles of conservation of momentum. Momentum is a crucial concept when objects interact. It is essentially the product of an object's mass and its velocity, representing how much "push" it has while moving. In isolated systems like our problem, momentum is conserved unless acted upon by an external force. This means the total momentum before and after the collision remains the same.
In this exercise, initially, only the dart has momentum. We calculate it using the formula \( p = m v \), where \( m \) is the mass and \( v \) is the velocity. After the dart hits and embeds into the lead sphere, both objects move together, combining their masses and sharing a new velocity, \( v_f \). Thus, immediately after the collision, the combined momentum (dart plus sphere) equals the initial momentum of the dart.
The equation for momentum conservation here is:

• Before collision: \( m_d v_d \)
• After collision: \((m_d + m_s) v_f \)

This ensures we can solve for the initial speed of the dart by comparing these two momentum values.

Circular motion is the movement of an object along the circumference of a circle or rotation along a circular path. In this exercise, the lead sphere and dart, after collision, need to swing in a complete circle.
The critical condition here is that at the top of their swing, the sphere-dart system has just enough velocity to overcome gravitational pull without tension in the wire. This is where we use the concept of centripetal force, which is necessary for circular motion.
For the sphere to perform a complete circle, the centripetal force at the top must equal the gravitational force. This balance is given by the equation:

• \( \frac{(m_d + m_s) v_{top}^2}{R} = (m_d + m_s)g \)

Here, \( v_{top} \) denotes the velocity at the top of the circle. The equation shows that this velocity depends on both the gravitational pull \( g \) and the radius \( R \) of the swing. By isolating and solving for \( v_{top} \), we can find the minimum speed necessary for a complete circular motion.

Energy conservation is a fundamental principle in physics, stating that energy cannot be created or destroyed, only transformed from one form to another. In the context of this problem, energy transforms between kinetic (motion) and potential (position) energy as the system oscillates.After the collision, we apply conservation of energy principles to understand how the motion transforms. At the lowest point of the swing (immediately post-collision), the system has maximum kinetic energy. As it ascends, kinetic energy converts to potential energy until, at the top, potential energy peaks while kinetic energy dips.
The conservation equation that captures this transformation as the system rises to complete the loop is:

• \( \frac{1}{2}(m_d + m_s) v_f^2 = \frac{1}{2}(m_d + m_s) v_{top}^2 + (m_d + m_s)g \times 2R \)

This provides a relationship between initial velocity \( v_f \) and the velocity at the top \( v_{top} \). By rearranging and solving, we see how much kinetic energy must be retained or introduced to keep the system moving in a full loop. Understanding these energy transformations is crucial to solving the problem and predicting motion correctly.