In Physics, there is often more than one way to solve a problem. Sometimes the t

Question

In Physics, there is often more than one way to solve a problem. Sometimes the trick to finishing your homework in less than the timescale for our Sun to turn into a Planetary Nebula (or more immediately, in time to watch the Monday night football game!) is to find the most efficient way to solve a problem. When working with simple harmonic oscillators it is important to remember than conservation of energy is an important tool in your kit! You have an oscillating 0.5 kg mass attached to a horizontal spring whose motion can be described by x(t) = 20 cos(8t), where x is in mm and t is in seconds. You want to know how fast it is moving when x = 10 mm?

A. Find a solution while using v(t) = d dt x(t) and no energy.

B. Find a solution while using E = K + U, but not v(t) = d dt x(t).

C. Now choose your method of solution: You have a 0.25 kg block undergoing SHM on the end of a horizontal spring with spring constant k = 250 N/m. When the block is 0.025 m from its equilibrium point you measure the velocity to be 0.5 m/s. What is x(t)?

Explanation / Answer

angular frequency=8*pi rad/sec

as we know, if spring constant is k, then angular frequency=sqrt(k/m)

==>8*pi=sqrt(k/0.5)

==>k=315.83 N/m

v(t)=dx/dt=-160*pi*sin(8*pi*t)

so when x=10 mm,

let time be t seconds

then 10=20*cos(8*pi*t)

==>t=41.667 ms

then at t=41.667 ms, v(t)=-160*pi*sin(8*pi*t)=-435.31 mm/s=0.43531 m/s

b)

at maximum displacement , speed=0

then total energy=potential energy of spring=0.5*k*0.02^2=0.063165 J

at x=10 mm, let the speed be v.

then total energy=0.5*k*0.01^2+0.5*0.5*v^2=0.063165

==>v=0.45353 m/s

hence result obtained from both the method are same.

c)energy method is utilised here

mass=m=0.25 kg

k=250 N/m

then angular frequency=sqrt(k/m)=31.623 rad/sec

let amplitude be A.

at amplitude, velocity=0

hence net energy=0.5*250*A^2

at 0.025 m, net energy=0.5*250*0.025^2+0.5*0.25*0.5^2=0.10938 J

equating net energy in both the cases,

we get A=0.02958 m

then x(t)=0.02958*cos(31.623*t)

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