The motion of the piston of an automobile engine is approximately simple harmoni

Question

The motion of the piston of an automobile engine is approximately simple harmonic. (Figure 1) If the stroke (twice the amplitude) of an engine is 0.100 m and the engine runs at 3500 rpm , compute the acceleration of the piston at the endpoint of its stroke. If the piston has a mass of 0.450 kg , what net force must be exerted on it at this point? What are the speed and kinetic energy of the piston at the midpoint of its stroke?

Part A

The problem suggests that you model the motion of the piston as simple harmonic. You are given enough information to determine the angular frequency of the system, ?, which is a basic physical property of the system, and the amplitude of oscillation, A, which is a quantity that describes the particular motion of the system occurring under the given conditions. What are ?and A?

Express the angular frequency in radians per second and the amplitude in meters. Separate your answers with a comma.

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SOLVE

Now that you've set up the problem, choose appropriate equations and solve for your unknowns.

Part B

Compute the acceleration a of the piston at the endpoint of its stroke.

Express your answer in meters per second squared.

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Part C

If the piston has a mass of 0.450 kg , what net force Fnet must be exerted on it at this point?

Express your answer in newtons.

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Part D

What are the speed v and kinetic energy K of the piston at the midpoint of its stroke?

Express the speed in meters per second and the energy in joules. Separate your answers with a comma.

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REFLECT

Think about whether your results make sense. If you change one of the given quantities, do the results change in a predictable way?

Part E

If the engine runs at 7000 rpm , what are the acceleration, net force, velocity, and kinetic energy? Use your equations from Parts B, C, and D.

Express acceleration in meters per second squared, force in newtons, speed in meters per second, and energy in joules. Separate yours answers with commas.

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Explanation / Answer

part A:

given that, engines sped is 3500 rpm

it means 3500 revolutions per minute

as 1 revolution=2*pi radians

==>3500 revolutions per minute=3500*2*pi/(60) radians/sec

=366.52 rad/sec

hence angular speed=w=366.52 rad/sec

given that stroke of the engine is 0.1 m

==>2*A=0.1

==>A=0.1/2=0.05 m

part B:
as we have assumed the motion to be simple harmonic,

let at any time displacement, y(t)=A*sin(w*t)

then veloicty=dy/dt=A*w*cos(w*t)

acceleration=dv/dt=-A*w^2*sin(w*t)

let at time t, the engine reaches end point of stroke point

==>y(t)=A
==>sin(w*t)=1

hence acceleration=-A*w^2*1=-A*w^2

magnitude of acceleration=A*w^2=0.05*366.52^2=6716.845 m/s^2

part C:

force=mass*acceleration

=0.45*6716.845=3022.58 N

part D:
at mid point, y(t)=0

==>sin(w*t)=0

==>cos(w*t)=sqrt(1-sin^2(w*t))=1

then speed=A*w*1=A*w=0.05*366.52=18.326 m/s

kinetic energy=0.5*mass*speed^2

=0.5*0.45*18.326^2=75.564 J

note: as each quanity is dependent upon A or w, if you change A or w or both, all the other dependent quantities will also change.

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