a) (8 points) Consider a mass of 2 kg on a frictionless horizontal plane, the mass is attached to a spring of spring constant k = 200 N/m and performing a simple harmonic motion. You are told that the maximum velocity of the mass is 20 m/s and that at t = 0 the velocity is -20 m/s. Assuming a solution of the kind in Equation (1) above, find A, w and oo for this harmonic motion.

a) (8 points) Consider a mass of 2 kg on a frictionless horizontal plane, the mass is attached to a spring of spring constant k = 200 N/m and performing a simple harmonic motion. You are told that the maximum velocity of the mass is 20 m/s and that at t = 0 the velocity is -20 m/s. Assuming a solution of the kind in Equation (1) above, find A, w and oo for this harmonic motion.

Name: The equation of motion for a simple harmonic oscillator (SHO) is:= -k
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Description: The equation of motion for a simple harmonic oscillator (SHO) is:= -krdt2where m is the mass and k the spring constant. A generic solution of the above differential equation canbe written in the form:x(t) = A cos(wt + ¢o).(1)Where w = Vk/m and A and

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Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

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Concept explainers

Simple harmonic motion

Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.

Simple Pendulum

A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.

Oscillation

In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.

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Question

The equation of motion for a simple harmonic oscillator (SHO) is:
= -kr
dt2
where m is the mass and k the spring constant. A generic solution of the above differential equation can
be written in the form:
x(t) = A cos(wt + ¢o).
(1)
Where w = Vk/m and A and øo are arbitrary constants to be determined by the initial conditions of
the motion.
a) (8 points) Consider a mass of 2 kg on a frictionless horizontal plane, the mass is attached to a spring
of spring constant k = 200 N/m and performing a simple harmonic motion. You are told that the
maximum velocity of the mass is 20 m/s and that at t = 0 the velocity is -20 m/s. Assuming a
solution of the kind in Equation (1) above, find A, w and o for this harmonic motion.

b) (3 points) Using your results from part a) write an expression for the velocity v(t) as a function of
time for the mass.
c) (3 points) Make a sketch of your function v(t) in part b) for any time t 2 0.
d) (2 points) Make a sketch of the kinetic energy of the mass for any t > 0.
e) (4 points) What can you say about the potential energy of the mass at time t = 0? Explain your
answer.

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