I do not understand how to slove these questions, please show the steps that you

Question

I do not understand how to slove these questions, please show the steps that you use in the process.

Part A.Consider the oscillation of a mass-spring system, where x = A cos(w t + theta). At the time t = 0, the mass is at x = 0 and it is moving to the right with a speed v0. Find the phase angle theata. Consider x as the projection of a counter-clockwise uniform circular motion.

1. theta = 3 pi/ 4

2. theta = pi /2

3. theta = 7 pi /4

4. theta = 5 pi/4

5. theta = 3 pi/ 2

6. theta = 0

7. theta = pi/3

8. theta = pi/4

9. theta = 2 pi

10. theta = pi

Part B.Find the total energy of the oscillation at t = T/8, where T is the period.

1. E = m v^2

2. E = 5/2 m v^2

3. E = 5/4 m v^2

4. E = 3/2mv^2

5. E = 1/4mv^2

6. E = 1/2mv^2

7. E = 3/4mv2

8. E = 4mv^2

9. E = 2 m v^2

10. E = 3 m v^2

Explanation / Answer

x = A cos(w t + theta)

when t = 0, x = 0

So, 0 = A cos(w*0 + theta)

So, 0 = A cos( theta )

So, cos( theta ) = 0

So, theta = acos(0) = pi/2 <--------answer

b)

Now, x = A*cos(Wt + theta)

So, speed, v = dx/dt = -A*W*sin(Wt + theta)

where AW = vo = initial speed

So, at t = T/16,

v = -AW*sin(W*(T/16) + pi/2)

at t = T/8,

v = vo*(sin(W*(T/8)) + pi/2)

we know, W = 2*pi/T

So, v = vo*(sin(2*pi/8) + pi/2) = vo*sin(pi/4 + pi/2) =vo*sin(3pi/4)

So, v = vo/sqrt(2)

So, total energy at t = T/16 s

E = 0.5*(m)v^2 = 0.5*m*(vo/sqrt(2))^2 = m*vo^2 <-----------answer

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