Question

A 4.00-kg object is attached to a spring and placed on frictionless, horizontal surface. A horizontal force of 27.0 N is required to hold the object at rest when it is pulled 0.200 m from its equilibrium position (the origin of the x axis). The object is now released from rest from this stretched position, and it subsequently undergoes simple harmonic oscillations.

(a) Find the force constant of the spring.
N/m
(b) Find the frequency of the oscillations.
Hz
(c) Find the maximum speed of the object.
m/s
(d) Where does this maximum speed occur?
x = ± m
(e) Find the maximum acceleration of the object.
m/s2
(f) Where does the maximum acceleration occur?
x = ± m
(g) Find the total energy of the oscillating system.
J
(h) Find the speed of the object when its position is equal to one-third of the maximum value.
m/s
(i) Find the magnitude of the acceleration of the object when its position is equal to one-third of the maximum value.
m/s^2

Answer 1

a)

`135`Nm⁻¹

b)

`0.925` Hz

c)

`1.2`ms⁻¹

d)

`0` m

e)

`6.7`ms⁻²

f)

`\pm 0.2` m

Step-by-step explanation:

a)

`F` = force required to hold the object at rest connected with stretched spring = 27 N

`x` = stretch in the spring from equilibrium position = 0.2 m

`k` = force constant of the spring

force required to hold the object at rest is same as the spring force , hence

`F = k x`

`k = (F)/(x)`

inserting the values

`k = (27)/(0.2) = 135` Nm⁻¹

b)

frequency of the oscillations is given as

`f =(1)/(2\pi )\sqrt{(k)/(m)}`

inserting the values

`f =(1)/(2(3.14) )\sqrt{(135)/(4)}\\f = 0.925` Hz

c)

`A` = Amplitude of oscillations = 0.2 m

`w` = angular frequency

Angular frequency is given as

`w = 2\pi f = 2 (3.14) (0.925) = 5.8` rads⁻¹

Maximum speed of oscillation is given as

`v_(max) = Aw`

`v_(max) = (0.2)(5.8)\\v_(max) = 1.2` ms⁻¹

d)

maximum speed of the object occurs at the equilibrium position, hence

`x = 0` m

e)

Maximum acceleration of oscillation is given as

`a_(max) = Aw^(2)`

`a_(max) = (0.2)(5.8)^(2)\\a_(max) = 6.7`ms⁻²

f)

maximum acceleration occurs when the object is at extreme positions, hence

`x = \pm 0.2` m