Problem 75 - College Physics

Question

\(\bullet\) A 20.0 -kg lead sphere is hanging from a hook by a thin wire 3.50 m long, and is free to swing in a complete circle. Suddenly it is struck horizontally by a 5.00 -kg steel dart that embeds itself in the lead sphere. What must be the minimum initial speed of the dart so that the combination makes a com- plete circular loop after the collision?

Step-by-Step Solution

Answer

The minimum initial speed of the dart is approximately 29.4 m/s.

Step 1: Determine the Conservation of Momentum Equation

When the dart hits the sphere, the system undergoes an inelastic collision because the dart embeds itself into the sphere. The principle of conservation of linear momentum applies here. Before the collision, only the dart has momentum. After the collision, the combined mass moves together. To express this, we use:\[ m_d \cdot v_d = (m_s + m_d) \cdot v_{after} \]where \( m_d = 5.00\, \text{kg} \) is the mass of the dart, \( v_d \) is the speed of the dart, \( m_s = 20.0\, \text{kg} \) is the mass of the sphere, and \( v_{after} \) is the speed of the dart-sphere system immediately after the collision.

Step 2: Calculate Post-Collision Velocity

Rearrange the momentum equation to solve for the velocity of the system immediately after the collision:\[ v_{after} = \frac{m_d \cdot v_d}{m_s + m_d} \]This will be used later once we know the dart's initial velocity needed for the complete swing.

Step 3: Analyze Energy Requirements for Complete Swing

For the combined dart-sphere system to complete a circular loop, it needs sufficient energy at the apex of the swing, which is 2 times the length of the wire above the starting point. At the top of the swing, the minimum speed needed at the top for circular motion is determined by gravitation:\[ v_{top} = \sqrt{g \cdot L} \]From mechanical energy conservation, energy at the bottom equals energy at the top plus kinetic energy needed for looping. Using:\[ \frac{1}{2} (m_s + m_d) v_{after}^2 = (m_s + m_d) g L + \frac{1}{2} (m_s + m_d) v_{top}^2 \]

Step 4: Solve for Minimum Initial Velocity of the Dart

Substitute the expression of \( v_{top} \) and \( v_{after} \) into the energy equation:\[ \frac{1}{2} (m_s + m_d) \left( \frac{m_d \cdot v_d}{m_s + m_d} \right)^2 = (m_s + m_d) g L + \frac{1}{2} (m_s + m_d) g L \]Solve for \( v_d \) to find the minimum initial speed of the dart needed for the complete loop. Keep in mind:\[ L = 3.50\, \text{m}; \quad g = 9.81\, \text{m/s}^2 \]Simplify and solve for \( v_d \).

Step 5: Final Calculation and Answer

Plug in the values and simplify the expressions. Solving for \( v_d \) gives:\[ v_d = \sqrt{2gL\left(1+\frac{m_s+m_d}{m_s}\right)} \]Compute \( v_d \) using the given masses and the value of \( g \). You will find that:\[ v_d \approx 29.4\, \text{m/s} \]

Key Concepts

Inelastic Collision

An inelastic collision is a type of collision where the colliding objects stick together after the impact. In the given problem, the momentum of the dart is transferred to the lead sphere as the dart embeds itself in the sphere. This means that both objects move together as one mass post-collision. During an inelastic collision, kinetic energy is not conserved, but the total momentum of the system before and after the impact is conserved. This principle can be mathematically expressed as:

• The initial momentum of the dart: \( m_d \cdot v_d \).
• Total momentum after collision: \((m_s + m_d) \cdot v_{after} \).

This equation shows that the momentum the dart brings into the collision is distributed between the dart and the lead sphere after the collision.

Mechanical Energy Conservation

Mechanical energy conservation is crucial when analyzing the motion of the dart-sphere system after the collision. Even though kinetic energy is not conserved during an inelastic collision, it becomes relevant when considering the system's swing. The mechanical energy at the bottom of the swing should be equal to the energy at the top, plus the kinetic energy required for looping. This can be expressed as:

• Energy at the bottom: \( \frac{1}{2} (m_s + m_d) v_{after}^2 \).
• Energy at the top: \((m_s + m_d) g L \).
• Kinetic energy required at the top for circular motion: \( \frac{1}{2} (m_s + m_d) v_{top}^2 \).

Understanding this energy conservation helps us calculate the initial speed needed for the dart, ensuring the combined system swings through a complete circular loop.

Circular Motion

In physics, circular motion involves any object moving along a circular path. For the dart-sphere system to complete a loop, it needs to maintain a minimum speed at the top of the circle. This required speed ensures that centripetal force can counteract the weight of the system due to gravity. The minimum speed at the top, derived from the equation for centripetal force in terms of gravitational forces, is calculated as:

• \( v_{top} = \sqrt{g \cdot L} \)

This concept confirms that at the apex of the swing, some kinetic energy must exist for the system not to fall due to gravity. Hence, circular motion principles help ensure that motion continues smoothly through a full loop.

Physics Problem Solving

Approaching physics problems requires logical analysis of underlying principles. For this exercise, breaking down the problem starts with understanding conservation principles. First, identify the inelastic collision to find how velocities combine, relying on momentum conservation. Then, address the energy aspects by analyzing mechanical energy conservation to ensure enough energy is present to complete a circle. Use the calculated post-collision velocity \( v_{after} \) to determine the energy requirements. Finally, bring in circular motion physics to confirm the minimum speed needed at the top of the swing. This ordered approach highlights the systematic techniques in tackling a complex physics problem, ensuring all aspects are covered methodically.