A block with mass m =6.5 kg is hung from a vertical spring. When the mass hangs

Question

A block with mass m =6.5 kg is hung from a vertical spring. When the mass hangs in equilibrium, the spring stretches x = 0.24 m. While at this equilibrium position, the mass is then given an initial push downward at v = 4.2 m/s. The block oscillates on the spring without friction.

1)

What is the spring constant of the spring?

N/m

2)

What is the oscillation frequency?

Hz

3)

After t = 0.41 s what is the speed of the block?

m/s

4)

What is the magnitude of the maximum acceleration of the block?

m/s2

5)

At t = 0.41 s what is the magnitude of the net force on the block?

N

6)

Where is the potential energy of the system the greatest?

At the highest point of the oscillation.

At the new equilibrium position of the oscillation.

At the lowest point of the oscillation.

PLEASE, CLEARLY WORK IT OUT CORRECTLY..THNKS

Explanation / Answer

given that

m =6.5 kg

v = 4.2 m/s.

the spring stretches x = 0.24 m

part(1)

we know that

k*x = m*g

k = m*g/x

k = 6.5*9.8 / 0.24

k = 265.41 N/m

part(2)

f = sqrt(k/m)/2*pi

f = sqrt( 265.41 / 6.5) / 2*3.14

f = 1.01 Hz

part(3)

t = 0.41 s

w = 2*pi*f

w = 2*3.14*1.01 = 6.38 rad/s

x(t) = A* cos(w*t-phi)

v(t) = dx(t)/dt

Our stating point is pi/2, going down from the equilibrium position.

Now we can use the velocity equation

v(t) = - A *w* sin (w*t - phi)    .......           (eq 1)

use conservation of energy,

(1/2) k x^2 = (1/2) m v^2

x^2 = v^2 * m/k

x = sqrt (v^2 * m / k) = sqrt(4.2*4.2*6.5/265.41)

x = A = 0.65 m

Now put all values in eq 1
v = -A*w *sin(w*t-pi/2)

v = - (0.65 m) * (6.38 ) sin (6.38 *0.41 - pi/2)

v = -4.33 m/s

part(5)

when x = A the sum of the forces = the spring force = kA

k*A = m*a

a = k*A /m

a = 265.41*0.65 /6.5

a = 26.54 m/s^2

part(5)

Fnet = k * x,

so we just need x when t = 0.41 sec.

x(t) = A* cos (w*t - phi)

x = (0.65) cos (6.38 *0.41 - pi/2)

x = 0.67 m

Fnet = k * x = 265.41*0.67 = 180.41 N

part(6)

the greatest potential energy of the system at the highest point of the oscillation.

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