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17. Periodic Motion

Energy in Simple Harmonic Motion

Physics

17. Periodic Motion

Energy in Simple Harmonic Motion

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4:54 minutes

Problem 31c

Textbook Question

Draw a graph like Fig. 14–11 for a horizontal spring whose spring constant is 95 N/m and which has a mass of 75 g on the end of it. Assume the spring was started with an initial amplitude of 2.0 cm. Neglect the mass of the spring and any friction with the horizontal surface. Use your graph to estimate the speed of the mass, for 𝓍 = 1.5 cm.

Verified step by step guidance

Understand the problem: The system described is a horizontal spring-mass system undergoing simple harmonic motion (SHM). The spring constant is 95 N/m, the mass is 75 g (0.075 kg), and the initial amplitude is 2.0 cm (0.02 m). We are tasked with estimating the speed of the mass when the displacement 𝓍 = 1.5 cm (0.015 m).

Step 1: Write the equation for the total energy in SHM. The total mechanical energy (E) in SHM is constant and is given by:

E = \frac{1}{2} k A^{2}

where k is the spring constant, and A is the amplitude of motion.

Step 2: Write the expression for the energy at any displacement 𝓍. The total energy is the sum of potential energy (U) and kinetic energy (K):

E = \frac{1}{2} k x^{2} + \frac{1}{2} m v^{2}

where x is the displacement, m is the mass, and v is the speed of the mass.

Step 3: Rearrange the energy equation to solve for the speed v. Subtract the potential energy term from the total energy to isolate the kinetic energy term, then solve for v:

v^{2} = \frac{1}{m} (\frac{1}{2} k A^{2} - \frac{1}{2} k x^{2})

Take the square root to find v:

v = \sqrt{\frac{1}{m} (\frac{1}{2} k A^{2} - \frac{1}{2} k x^{2})}

Step 4: Substitute the known values into the equation. Use k = 95 N/m, m = 0.075 kg, A = 0.02 m, and x = 0.015 m. Simplify the terms inside the square root to calculate the speed v. Note that you do not need to compute the final value here, but ensure all units are consistent (meters, kilograms, seconds).

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Key Concepts

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

Hooke's Law

Hooke's Law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position, expressed as F = -kx, where k is the spring constant and x is the displacement. This principle is fundamental in understanding how springs behave under various loads and is essential for analyzing the motion of the mass attached to the spring.

Simple Harmonic Motion (SHM)

Simple Harmonic Motion is a type of periodic motion where an object oscillates around an equilibrium position. In the case of a mass-spring system, the motion is characterized by a restoring force proportional to the displacement, leading to sinusoidal motion. Understanding SHM is crucial for predicting the behavior of the mass as it moves back and forth along the spring.

Energy Conservation in Oscillatory Systems

In oscillatory systems like a mass-spring setup, mechanical energy is conserved, alternating between potential energy stored in the spring and kinetic energy of the moving mass. At maximum displacement, potential energy is at its peak, while kinetic energy is zero, and vice versa at the equilibrium position. This concept is vital for calculating the speed of the mass at any given displacement.

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Textbook Question

Draw a graph like Fig. 14–11 for a horizontal spring whose spring constant is 95 N/m and which has a mass of 75 g on the end of it. Assume the spring was started with an initial amplitude of 2.0 cm. Neglect the mass of the spring and any friction with the horizontal surface. Use your graph to estimate the potential energy for 𝓍 = 1.5 cm.

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Textbook Question

FIGURE EX15.7 is the position-versus-time graph of a particle in simple harmonic motion. What is v_max?

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Textbook Question