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

Answer:

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

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

$F = k x$

$k = \frac{F}{x}$

inserting the values

$k = \frac{27}{0.2} = 135$ Nm⁻¹

b)

frequency of the oscillations is given as

$f =\frac{1}{2\pi }\sqrt{\frac{k}{m}}$

inserting the values

$f =\frac{1}{2(3.14) }\sqrt{\frac{135}{4}}\\f = 0.925$ 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