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.

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Solution

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:

Where, T is the time period

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 =

A = Maximum displacement

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

(iii) Maximum velocity, v_max = Aω

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

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Similar questions

Q. A spring having with a spring constant 1200 N m–1 is mounted on a horizontal table as shown in Fig. 14.24. A mass of 3 kgis 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.

Q. A spring having a spring constant

1200 {Nm}^{- 1}

is mounted on a horizontal table as shown. A mass of

3.0 kg

is attached to the free end of the spring. The mass is then pulled sideways to a distance of

0.2 cm

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

Exercise:

[ A spring with a spring constant of 1200 N m

^{- 1}

is mounted on a horizontal table as shown in the Figure. 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. ]

In Exercise, let us take the position of mass when the spring is unstretched as x = 0, and the direction from left to right as the positive direction of x-axis. Give x as a function of time t for the oscillating mass if at the moment we start the stopwatch (t = 0), the mass is
(a) at the mean position,
(b) at the maximum stretched position, and
(c) at the maximum compressed position.
In what way do these function for SHM differ from each other in frequency, in amplitude or the initial phase?

Q. Figure (a) attached shows a spring of force constant k clamped rigidly at one end and a mass m attached to its free end. A force F applied at the free end stretches the spring. Figure (b) shows the same spring with both ends free and attached to a mass m at either end. Each end of the spring in Fig. (b) is stretched by the same force F.
(a) What is the maximum extension of the spring in the two cases ?
(b) If the mass in Fig (a) and the two masses in Fig (b) are released, what is the period of oscillation in each case ?

Figure 14.30 (a) shows a spring of force constant k clamped rigidly at one end and a mass m attached to its free end. A force F applied at the free end stretches the spring. Figure 14.30 (b) shows the same spring with both ends free and attached to a mass m at either end. Each end of the spring in Fig. 14.30(b) is stretched by the same force F.

(a) What is the maximum extension of the spring in the two cases?

(b) If the mass in Fig. (a) and the two masses in Fig. (b) are released, what is the period of oscillation in each case?

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