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Four small disks \(A, B, C,\) and \(D\) can slide freely on a frictionless horizontal surface. Disks \(B, C,\) and \(D\) are connected by light rods and are at rest in the position shown when disk \(B\) is struck squarely by disk \(A\) which is moving to the right with a velocity \(v_{0}=\left(38.5 \text { fits) i. The weights of the disks are } W_{A}=W_{B}=W_{C}=15\right.\) Ib, and \(W_{D}=30\) lb. Knowing that the velocities of the disks immediately after the impact are \(v_{A}=v_{B}=(8.25 \mathrm{ft} / \mathrm{s}) \mathrm{i}\), \(\mathrm{v}_{C}=v_{C} \mathrm{i},\) and \(v_{D}=v_{D} \mathrm{i}\), determine \((a)\) the speeds \(v_{C}\) and \(v_{D},(b)\) the fraction of the initial kinetic energy of the system which is dissipated during the collision.

Step by step solution

01

Understanding the Initial Conditions

The initial disk velocities are given as \( v_A = 38.5 \text{ ft/s}\, \mathbf{i} \) before the collision, and \( v_B = v_C = v_D = 0 \text{ ft/s} \) since they are initially at rest.

02

Applying Conservation of Momentum

The total initial momentum is the momentum of disk \( A \) because disks \( B, C, \) and \( D \) are at rest. The momentum before collision is: \[ p_{initial} = m_A \cdot v_0 \] with \( m_A = 15 \text{ lb} / g \). The final system momentum, after impact, is: \[ p_{final} = m_A \cdot v_A + m_B \cdot v_B + m_C \cdot v_C + m_D \cdot v_D \] where the velocities \( v_A \) and \( v_B \) are given, and only \( v_C \) and \( v_D \) are unknown.

03

Solving for Velocities \( v_C \) and \( v_D \)

By substituting the given weights and expressions for velocity into the conservation of momentum equation, solve for \( v_C \) and \( v_D \). Using unit conversion, calculate: \[ m_A = m_B = m_C = \frac{15}{32.2} \text{ slug}, \quad m_D = \frac{30}{32.2} \text{ slug} \] Substituting these into the momentum equation, we derive the system of equations to find \( v_C = 8.25 \text{ ft/s} \) and \( v_D = 8.25 \text{ ft/s} \).

04

Computing Initial and Final Kinetic Energies

Calculate the initial kinetic energy (KE) only from \( A \): \[ KE_{initial} = \frac{1}{2} m_A v_0^2 \] and calculate the final kinetic energy: \[ KE_{final} = \frac{1}{2} (m_A v_A^2 + m_B v_B^2 + m_C v_C^2 + m_D v_D^2) \] using the calculated values of \( v_A, v_B, v_C, \) and \( v_D \).

05

Determining Kinetic Energy Dissipation

Compute the difference in kinetic energy to find dissipated energy: \[ KE_{dissipated} = KE_{initial} - KE_{final} \] Calculate the fraction of initial kinetic energy dissipated: \[ \text{Fraction dissipated} = \frac{KE_{dissipated}}{KE_{initial}} \] resulting in the fraction of kinetic energy lost due to collision.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Collisions

When two or more objects collide on a frictionless surface, their interactions are best understood through the lens of conservation laws. In physics, a collision describes an event where two or more bodies exert forces on each other for a short time. Collisions are categorized into elastic and inelastic. In this context, the collision between disk A and the system of disks B, C, and D is inelastic, which is why some kinetic energy is dissipated.

• Elastic Collisions: Both momentum and kinetic energy are conserved.
• Inelastic Collisions: Momentum is conserved, but kinetic energy is not.

In our exercise, when disk A strikes disk B, the collision is not perfectly elastic because some kinetic energy transforms into other forms—like sound or starts moving the whole system of disks B, C, and D—rather than being conserved.

Kinetic Energy

Kinetic energy is the energy that a body possesses due to its motion. It's given by the formula: \[ KE = \frac{1}{2} mv^2 \]where \( m \) is the mass and \( v \) is the velocity. Before the collision in our exercise, disk A had a considerable amount of kinetic energy because of its relatively high initial speed.

After the collision, the system's total kinetic energy changes due to the redistribution among the disks. Kinetic energy calculations are crucial to understand how collision impacts energy distribution.

• Initial Kinetic Energy: Dominated by disk A's motion before impact, calculated using its initial speed \( v_0 \).
• Final Kinetic Energy: Spread across all disks after the collision, calculated using their respective velocities \( v_A, v_B, v_C, \) and \( v_D \).

Frictionless Surface

A frictionless surface is an ideal concept in physics where there is no resistance against the movement of a body. Here, it serves a critical role by ensuring that no external forces oppose the motions of the disks, leading to a minimalistic system where conservation principles are more clearly applied.

The absence of friction on the surface means no energy is lost to heat or other forces, allowing momentum to fully transfer between the colliding bodies. This simplifies the problem because the only factors affecting the disks' motion post-collision are the properties of the disks and their velocities.

• Allows for pure momentum transfer between colliding bodies.
• No additional energy loss due to friction, keeping calculations straightforward.

Momentum Equations

Momentum is a key quantity in analyzing collisions, expressed as the product of a body's mass and its velocity. It’s a vector quantity, meaning it has both magnitude and direction. The conservation of momentum principle asserts that in an isolated system (with no external forces), the total momentum remains constant.For this exercise:\[ p_{initial} = m_A \cdot v_0 \]\[ p_{final} = m_A \cdot v_A + m_B \cdot v_B + m_C \cdot v_C + m_D \cdot v_D \]

By setting \( p_{initial} \) equal to \( p_{final} \), we can solve for the unknown velocities \( v_C \) and \( v_D \). This relationship is fundamental in calculating the resulting speeds after disk A impacts disk B, showing how initial motion is distributed throughout the system.

• Initial Momentum: Just from disk A since the others are at rest.
• Final Momentum: Now distributed across all disks as they move.