5. The plot of the potential energy of a mass m moving along the x axis is shown

Question

5. The plot of the potential energy of a mass m moving along the x axis is shown above. If the total mechanical energy of the particle is E,

a)  Find both the left and right amplitudes as a function of E

b) Write down an expression for the mechanical energy of the particle, and find the period of oscillation of the particle.

The plot of the potential energy of a mass m moving along the x axis is shown above. If the total mechanical energy of the particle is E, Find both the left and right amplitudes as a function of E Write down an expression for the mechanical energy of the particle, and find the period of oscillation of the particle.

Explanation / Answer

5a)

total mechanical energy E =0.5mv^2 +mgx

right side amplitude at x =A velocity is zero

so total mechanical energy E =0+mgA

Amplitude A =E/mg

left side amplitude

at x=A vellocity is zero

so E =0+1/2 kA^2

A^2 =2E/k

Amplitude A =sqrt(2E/k)

b)

for left side

total mechanical energy E =0.5mv^2 +0.5kx^2

time peroid T =2pi(m/k)^0.5

for right side

E=mgx+0.5mv^2

time peroid T =2pi(l/g)^0.5

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