Question

A spring having with a spring constant 1200 N m^–1 is mounted on a horizontal table as shown in Fig. A mass of 3 kg is attached to the free end of the spring. The mass is then pulled sideways to a distance of 2.0 cm and released.

Determine (i) the frequency of oscillations, (ii) maximum acceleration of the mass, and (iii) the maximum speed of the mass.

Answer 1

Spring constant, k = 1200 N m^–1

Mass, m = 3 kg

Displacement, A = 2.0 cm = 0.02 cm

(i) Frequency of oscillation v, is given by the relation:

`v = 1/T = 1/(2pi) sqrt(k/m)`

Where, T is the time period

`:. v = 1/(2xx3.14) sqrt(1200/3) = 3.18 "m/s"`

Hence, the frequency of oscillations is 3.18 cycles per second.

ii) Maximum acceleration (a) is given by the relation:

a = ω² A

Where

ω = Angular frequency  = `sqrt(k/m)`

A = Maximum displacement

`:. a = k/m A = (1200xx0.02)/(3) =  8 ms^(-2)`

Hence, the maximum acceleration of the mass is 8.0 m/s²

iii) Maximum velocity, v_max = Aω

`= A sqrt(k/m) = 0.02 xx sqrt(1200/3) = 0.4 "m/s"`

Hence, the maximum velocity of the mass is 0.4 m/s.

Answer 2

K = 1200 `Mn^(-1)`; m = 3.0 kg, a= 2.0 cm = 0.02 m

i) Frequency, `v =  1/T = 1/(2pi) sqrt(k/m) = 1/(2xx3.14) sqrt(1200/3) = 3.2 s^(-1)`

ii) Acceleration, A = `omega^2` `" " y = k/m  y`

Acceleration will be maximum when y is maximum i.e y = q

:. max acceleration,` A_"max" = (ka)/m =(1200xx0.02)/3 = 8 ms^(-2)`

iii) Max speed of the mass will be when it is passing throught mean position

`V_"max" = aomega = sqrt(k/m) = 0.02 xx sqrt(1200/3) = 0.4 ms^(-1)`