Question

A block with mass m = 6.2 kg is attached to two springs with spring constants k_left = 31 N/m and k_right = 49 N/m. The block is pulled a distance x = 0.2 m to the left of its equilibrium position and released from rest.

1)

What is the magnitude of the net force on the block (the moment it is released)?

N

2)

What is the effective spring constant of the two springs?

N/m

3)

What is the period of oscillation of the block?

s

4)

How long does it take the block to return to equilibrium for the first time?

s

5)

What is the speed of the block as it passes through the equilibrium position?

m/s

6)

What is the magnitude of the acceleration of the block as it passes through equilibrium?

m/s²

7)

Where is the block located, relative to equilibrium, at a time 1.07 s after it is released? (if the block is left of equilibrium give the answer as a negative value; if the block is right of equilibrium give the answer as a positive value)

m

8)

What is the net force on the block at this time 1.07 s? (a negative force is to the left; a positive force is to the right)

N

9)

What is the total energy stored in the system?

J

10)

If the block had been given an initial push, how would the period of oscillation change?

the period would increase

the period would decrease

the period would not change

Answer 1

Given,

K(left) = 31 N/m ; K(right) = 49 N/m ; m = 6.2 kg ; x = 0.2 m

1)The net force on the block is the Hook's force.

F = -kx

F = [K(left) + K(right) ] x = - (31+49) x 0.2 = -16 N

Hence, F = -16 N (magnitude only)

2)The effective spring constant is:

keff = F/x = 16/0.2 = 80 Nm/

Hence, Keff = 80 N/m

3)The period of oscillation is:

T = 2pi sqrt (m/k)

T = 2 x 3.14 x sqrt (6.2/80) = 1.75 s

Hence, T = 1.75 s

4)let t be the time it take to reach back,

t = T/4 = 1.75/4 = 0.44 s

Hence, t = 0.44 s

5)from conservation of energy

1/2 k x^2 = 1/2 mv^2

v = x * sqrt (k/m)

v = 0.2 x sqrt (80/6.2) = 0.72 m/s

Hence, v = 0.72 m/s