Ch 15: Oscillations

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 15: Oscillations

Problem 51

Chapter 15, Problem 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?

Verified step by step guidance

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 \( v_{\text{dart}} \): \( v_{\text{dart}} = \frac{m_{\text{total}}v}{m_{\text{dart}}} \).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

12m

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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.

Recommended video:

Guided course

06:18

Energy in Horizontal Springs

Related Practice

Textbook Question

A mass hanging from a spring oscillates with a period of 0.35 s. Suppose the mass and spring are swung in a horizontal circle, with the free end of the spring at the pivot. What rotation frequency, in rpm, will cause the spring's length to stretch by 15%?

views

Textbook Question

A 200 g block hangs from a spring with spring constant 10 N/m. At t = 0 s the block is 20 cm below the equilibrium point and moving upward with a speed of 100 cm/s. What are the block's distance from equilibrium when the speed is 50 cm/s?

Textbook Question

Your lab instructor has asked you to measure a spring constant using a dynamic method—letting it oscillate—rather than a static method of stretching it. You and your lab partner suspend the spring from a hook, hang different masses on the lower end, and start them oscillating. One of you uses a meter stick to measure the amplitude, the other uses a stopwatch to time 10 oscillations. Your data are as follows: Use the best-fit line of an appropriate graph to determine the spring constant.

Textbook Question

A 200 g block hangs from a spring with spring constant 10 N/m. At t = 0 s the block is 20 cm below the equilibrium point and moving upward with a speed of 100 cm/s. What are the block's a. Oscillation frequency?

Textbook Question

A compact car has a mass of 1200 kg. Assume that the car has one spring on each wheel, that the springs are identical, and that the mass is equally distributed over the four springs. What will be the car's oscillation frequency while carrying four 70 kg passengers?

Textbook Question

Scientists are measuring the properties of a newly discovered elastic material. They create a 1.5-m-long, 1.6-mm-diameter cord, attach an 850 g mass to the lower end, then pull the mass down 2.5 mm and release it. Their high-speed video camera records 36 oscillations in 2.0 s. What is Young's modulus of the material?