2022HSCShort answer

Q15 (b)

3 marks

Question

The diagrams show two positions of a single piston in the cylinder chamber of a motorcycle. The piston moves vertically, in simple harmonic motion, between a maximum height of $0.17$ metres and a minimum height of $0.05$ metres.

The mass of the piston is $0.8\,\mathrm{kg}$ . The piston completes $40$ cycles per second.

What is the resultant force on the piston, in newtons, that produces the maximum acceleration of the piston? Give your answer correct to the nearest newton.

Uncropped source page 13 for question 15.b; requires human review and cropping — Reviewed visual · source page 13

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Solution

Criteria	Marks
Provides correct solution	3
Obtains an equation of motion of the piston, or equivalent merit	2
Finds the period of the motion, or equivalent merit	1

Sample answer:
$a=\frac{0.17-0.05}{2}=0.06\text{ m}$
$\text{Period of motion}=\frac1{40}=0.025\text{ s}$
Equation of motion simple harmonic motion (SHM) is
$\ddot x=-n^2(x-b)$
$n=\frac{2\pi}{T}=\frac{2\pi}{0.025}=80\pi$
Centre of motion $b=0.11$ .
$\therefore\ddot x=-(80\pi)^2(x-0.11)$
Acceleration is maximum at an extremity,
$\ddot x_{\max}=-(80\pi)^2(0.05-0.11)=384\pi^2\text{ m s}^{-2}$
The maximum force on the piston $f=m\ddot x$ :
$f=0.8\times384\pi^2\text{ newtons}\approx3032\text{ newtons}$

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Mechanics

Simple harmonic motionDerive and interpret displacement, velocity and acceleration models for simple harmonic motion.

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