Steven W. answered • 07/18/18

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Hi Nechama!

This questions looks like it is defining position as the "x" component of the piston's position as the wheel spins it around. That x position is described by a simple harmonic motion expression. This is demonstrated by the gif at the top of this page:

https://kaiserscience.wordpress.com/physics/waves/simple-harmonic-motion/shm-circular-motion-to-linear-motion-gif/

https://kaiserscience.wordpress.com/physics/waves/simple-harmonic-motion/shm-circular-motion-to-linear-motion-gif/

As such, we can attempt to describe the position function with either of the standard equations of motion for simple harmonic oscillation in one dimension:

x(t) = Asin(ωt + φ) or Acos(ωt + φ)

where

A = amplitude of the motion (maximum displacement from equilibrium)

ω = angular frequency of the motion (in radians per second (rad/s))

φ = phase shift (this is generally not needed if, at t = 0, the oscillating object is at either the equilibrium position or one of the amplitudes)

If the wheel has radius A, then A is the amplitude, the farthest the object can get from equilibrium in the x direction (as shown in the gif). Since the object is at x = A at t = 0, we will not need the phase shift term φ.

What we do need is the form of the standard equation which gives x = A at t = 0. The sin version cannot do this, since t = 0 would give us sin(0), which is 0. But cos can, since at t = 0, we get cos(0) = 1. Thus, if:

x(t) = Acos(ωt), then x(0) = A, as we require.

So we will take x(t) = Acos(ωt)

From the problem:

A = 0.250 m

ω = 12 rad/s

(a) To answer this part, we just evaluate the above expression at t = 1.15 s.

x(1.15 s) = (0.250 m)cos(12 rad/s * 1.15 s)

[note: since the argument of the cos function is in radians, you need to have your calculator in "radians" mode to get the correct result]

(b) and (c) There is an associated sequence of linear velocity and acceleration functions which follow from x(t) = Acos(ωt). They are:

v(t) = -ωAsin(ωt)

a(t) = -ω

^{2}Acos(ωt)(in calculus terms, these are just the first and second time derivatives of the position function)

To answer (b) and (c), just evaluate these expressions at t = 1.15 s. Remember to keep that calculator in radians! :)

I hope this helps. If you have any questions, please do not hesitate to ask. Best of luck!

I hope this helps. If you have any questions, please do not hesitate to ask. Best of luck!