Question

A particle undergoes simple harmonic motion (SHM) in one dimension. The r coordinate of the particle as a function of time is r(t)Aco() where A is the called the amptde" and w is called the "angular frequency." The motion is periodic with a period T given by Many physical systems are described by simple harmonic motion. Later in this course we will see, for example, that SHM describes the motion of a particle attached to an ideal spring. (a) What are the dimensions of the amplitude A and the angular frequency w? You may indicate the dimensions using mks units (b) Determine an expression for the velocity v(t). Explain why the units of your expression make sense, that is, why the coefficient that multiplies your sinusoidal function has the cur rent units. (c) Determine an expression for the acceleration a(t). Explain why the units of your expres- sion make sense, that is, why the coefficient that multiplies your sinusoidal function has the current units. (d) What is the average velocity between time0 and timeT/4? That is, calculate How does this average velocity compare to the maximum magnitude of the velocity during this time interval? (e) What is the average acceleration between time t- T/4 and timet- T/2? How does this average acceleration compare to the maximum magnitude of the acceleration during this time interval? (f) What is the displacement where the acceleration is a maximum? What is the displace- ment where the acceleration is a minimum? (g) What is the displacement where the speed is a maximum?

Answer 1

(a) Amplitude has dimensions of length and its mks unit is meter.

Frequency has dimension 1/T and its unit is sec^-1

(b) Velocity = $dx/dt = d(Acos\omega t)/dt$

$dx/dt = d(Acos\omega t)/dt = -A\omega sin\omega t$

as the unit of velocity is m/s, The coefficient A$\omega$ has the units of m/s as the sine function is dimensionless

(c) $Acceleration = d^2x/dt^2 =d^2(Acos\omega t)/dt^2 = -A\omega ^2cos\omega t$

as the unit of accleration is m/s², The coefficient A$\omega$² has the units of m/s² as the cos function is dimensionless