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A mass of 0.30 kg on the end of a spring oscillates with a period of...

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Physics

A mass of 0.30 kg on the end of a spring oscillates with aperiod of 0.45 s and an amplitude of 0.15 m . A) Find the velocitywhen it passes the equilibrium point. B) Find the total energy ofthe system. C) Find the spring constant. D) Find the maximumacceleration of the mass.

Answer & Explanation Solved by verified expert
4.3 Ratings (677 Votes)

let position at any time be given as

x(t)=A*sin(w*t)

where A=amplitude=0.15 m

w=angular frequency=2*pi/period=2*pi/0.45=13.963 rad/sec

hence x(t)=0.15*sin(13.963*t) m

veloicty=dx/dt=0.15*13.963*cos(13.963*t)=2.0944*cos(13.963*t) m/s

as angular frequency=sqrt(spring constant/mass)

==>13.963=sqrt(spring constant/0.3)

==>spring constant=13.963^2*0.3=58.49 N/m

a)

when the mass passes through equilibrium, x(t)=0

==>w*t=0

at that time, speed=2.0944*cos(0)=2.0944 m/s

part b:

as there is no friction, total energy of the system will be conserved.

at the highest displacement point, speed of the mass is 0.

then total energy of the system=potential energy of the spring=0.5*spring cosntant*amplitude^2

=0.5*58.49*0.15^2=0.658 J

part c:

spring constant is 58.49 N/m

part d:

acceleration of the mass=dv/dt=-2.0944*13.963*sin(13.963*t)

as maximum value of sine function is 1,maximum acceleration=2.0944*13.963=29.244 m/s^2
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