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Ch 15: Oscillations
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 15: OscillationsProblema 51
Capitolo 15, Problema 51

A 500 g wood block on a frictionless table is attached to a horizontal spring. A 50 g dart is shot into the face of the block opposite the spring, where it sticks. Afterward, the spring oscillates with a period of 1.5 s and an amplitude of 20 cm. How fast was the dart moving when it hit the block?

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Convert the given masses into kilograms: The mass of the wood block is 500 g = 0.5 kg, and the mass of the dart is 50 g = 0.05 kg.
Determine the total mass of the system after the dart sticks to the block: \( m_{\text{total}} = m_{\text{block}} + m_{\text{dart}} \).
Use the relationship between the period of oscillation and the spring constant: The period \( T \) is related to the spring constant \( k \) and the total mass \( m_{\text{total}} \) by the formula \( T = 2\pi \sqrt{\frac{m_{\text{total}}}{k}} \). Rearrange this equation to solve for \( k \): \( k = \frac{4\pi^2 m_{\text{total}}}{T^2} \).
Relate the amplitude and spring constant to the velocity of the block-dart system after the collision: The maximum potential energy stored in the spring is \( U = \frac{1}{2}kA^2 \), where \( A \) is the amplitude. This energy is equal to the kinetic energy of the block-dart system immediately after the collision: \( KE = \frac{1}{2}m_{\text{total}}v^2 \). Set \( U = KE \) and solve for \( v \): \( v = \sqrt{\frac{kA^2}{m_{\text{total}}}} \).
Use the principle of conservation of momentum to find the initial velocity of the dart: Before the collision, the dart has momentum \( p = m_{\text{dart}}v_{\text{dart}} \), and the block is stationary. After the collision, the total momentum is \( m_{\text{total}}v \), where \( v \) is the velocity of the block-dart system. Set the initial and final momenta equal: \( m_{\text{dart}}v_{\text{dart}} = m_{\text{total}}v \). Solve for \( v_{\text{dart}} \): \( v_{\text{dart}} = \frac{m_{\text{total}}v}{m_{\text{dart}}} \).

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Conservation of Momentum

The principle of conservation of momentum states that in a closed system, the total momentum before an event must equal the total momentum after the event. In this scenario, when the dart collides with the wood block, the momentum of the dart is transferred to the block, allowing us to calculate the dart's initial velocity using the combined mass and the final velocity of the system.
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Conservation Of Momentum

Simple Harmonic Motion (SHM)

Simple Harmonic Motion refers to the oscillatory motion of an object where the restoring force is directly proportional to the displacement from its equilibrium position. The period of oscillation, which is given in the question, can be used to determine the spring constant and the mass of the system, providing insights into the dynamics of the block-spring system after the collision.
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Simple Harmonic Motion of Pendulums

Kinetic Energy and Potential Energy in Springs

In a spring system, kinetic energy (KE) and potential energy (PE) interchange during oscillation. The potential energy stored in the spring at maximum displacement (amplitude) can be equated to the kinetic energy of the dart just before the collision, allowing us to analyze the energy transfer and calculate the dart's speed using the relationship between kinetic and potential energy.
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Energy in Horizontal Springs
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