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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
All textbooksKnight Calc 5th EditionCh 15: OscillationsProblem 49a
Chapter 15, Problem 49a

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?

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1
Convert the given quantities into SI units: The mass of the block is 200 g, which is equivalent to 0.2 kg. The spring constant is already in SI units (10 N/m). The displacement from equilibrium is 20 cm, which is 0.2 m, and the speed is 100 cm/s, which is 1 m/s.
Recall the formula for the angular frequency of a mass-spring system: \( \omega = \sqrt{\frac{k}{m}} \), where \( k \) is the spring constant and \( m \) is the mass of the block. Substitute the values of \( k = 10 \ \text{N/m} \) and \( m = 0.2 \ \text{kg} \) into the formula.
Calculate the oscillation frequency \( f \) using the relationship between angular frequency and frequency: \( f = \frac{\omega}{2\pi} \). Use the value of \( \omega \) obtained in the previous step.
Interpret the result: The oscillation frequency \( f \) represents the number of complete oscillations the block makes per second. Ensure the units of the final answer are in hertz (Hz).
Verify the assumptions: Confirm that the system is ideal (no damping or external forces) and that the spring follows Hooke's law, as these are necessary conditions for the formulas used to be valid.

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Key Concepts

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

Spring Constant

The spring constant, denoted as 'k', is a measure of a spring's stiffness. It quantifies the force required to stretch or compress the spring by a unit distance. In this case, a spring constant of 10 N/m means that a force of 10 Newtons is needed to stretch the spring by 1 meter. This property is crucial for understanding the dynamics of oscillatory motion.
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Oscillation Frequency

Oscillation frequency refers to the number of complete cycles of motion that occur in a unit of time, typically measured in Hertz (Hz). For a mass-spring system, the frequency can be calculated using the formula f = (1/2π)√(k/m), where 'k' is the spring constant and 'm' is the mass. This concept is essential for determining how quickly the block will oscillate around its equilibrium position.
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Equilibrium Position

The equilibrium position is the point at which the net force acting on the block is zero, meaning the spring is neither compressed nor stretched. In this scenario, the block's equilibrium position is where the gravitational force is balanced by the spring force. Understanding this position is vital for analyzing the motion of the block as it oscillates above and below this point.
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Related Practice
Textbook Question

An ultrasonic transducer, of the type used in medical ultrasound imaging, is a very thin disk (m = 0.10 g) driven back and forth in SHM at 1.0 MHz by an electromagnetic coil. What is the disk's maximum speed at this amplitude?

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Textbook Question

Astronauts in space cannot weigh themselves by standing on a bathroom scale. Instead, they determine their mass by oscillating on a large spring. Suppose an astronaut attaches one end of a large spring to her belt and the other end to a hook on the wall of the space capsule. A fellow astronaut then pulls her away from the wall and releases her. The spring's length as a function of time is shown in FIGURE P15.46. What is her speed when the spring's length is 1.2 m?

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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.

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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?

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Textbook Question

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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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?

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