A mass attached to a spring is free to oscillate, with angular ve… - Vidyadip

Question

A mass attached to a spring is free to oscillate, with angular velocity $\omega,$ in a horizontal plane without friction or damping. It is pulled to a distance x₀ and pushed towards the centre with a velocity $\upsilon_0$ at time t = 0. Determine the amplitude of the resulting oscillations in terms of the parameters $\omega,$ x₀ and $\upsilon_0.$ [Hint: Start with the equation $\text{x}=\text{a}\cos(\omega\text{t}+\theta)$ and note that the initial velocity is negative.]

✓

Answer

The displacement rquation for an oscillating mass is given by:

$\text{x}=\text{A}\cos(\omega\text{t}+\theta)$

where,

A is the amplitude

x is the displacement

$\theta$ is the phase constant

Velcoity, $\text{v}=\frac{\text{dx}}{\text{dt}}=-\text{A}\omega\sin(\omega\text{t}+\theta)$

At, t = 0, x = x₀

$\text{x}_0=\text{A}\cos\theta=\text{x}_0\ .....(\text{i})$

and, $\frac{\text{dx}}{\text{dt}}=-v_0=\text{A}\omega\sin\theta$

$\text{A}\sin\theta=\frac{v_0}{\omega}\ ....(\text{ii})$

Squaring and adding equations (i) and (ii), we get:

$\text{A}^2(\cos^2\theta+\sin^2\theta)=\text{x}_0^2+\Big(\frac{v_0^2}{\omega^2}\Big)$

$\therefore\ \text{A}=\sqrt{\text{x}_0^2+\Big(\frac{v_0}{\omega}\Big)^2}$

Hence, the amplitude of the resulting oscillation is $\text{x}_0^2+\Big(\frac{v_0}{\omega}\Big)^2.$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Oscillations→Oscillations — all question types→Physics STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

A vessel of volume V₀ contains an ideal gas at pressure P₀ and temperature T. Gas is continuously pumped out of this vessel at a constant volume-rate dV I dt = r keeping the temperature constant. The pressure of the gas being taken out equals the pressure inside the vessel. Find, (1) the pressure of the gas as a function of time. (2) the time taken before half the original gas is pumped out.

A pebble of mass 0.05kg is thrown vertically upwards. Give the direction and magnitude of the net force on the pebble,

During its upward motion
During its downward motion.
At the highest point where it is momentarily at rest. Do your answers change if the pebble was thrown at an angle of 45° with the horizontal direction? Ignore air resistance.

A closed cubical box is made of perfectly insulating material and the only way for heat to enter or leave the box is through two solid cylindrical metal plugs, each of cross-sectional area 12cm² and length 8cm fixed in the opposite walls of the box. The outer surface of one plug is kept at a temperature of 100°C while the outer surface of other plug is maintained at a temperature of 4°C. The thermal conductivity of the material of the plug is 2.0W/ m-°C. A source of energy generating 13W is enclosed inside the box. Find the equilibrium temperature of the inner surface of the box assuming that it is the same at all points on the inner surface.

One end of a long string of linear mass density 8.0 × 10^–3kg m^–1 is connected to an electrically driven tuning fork of frequency 256Hz. The other end passes over a pulley and is tied to a pan containing a mass of 90kg. The pulley end absorbs all the incoming energy so that reflected waves at this end have negligible amplitude. At t = 0, the left end (fork end) of the string x = 0 has zero transverse displacement (y = 0) and is moving along positive y-direction. The amplitude of the wave is 5.0cm. Write down the transverse displacement y as function of x and t that describes the wave on the string.

A man wants to reach from A to the opposite corner of the square C Fig. The sides of the square are 100m. A central square of 50m × 50m is filled with sand. Outside this square, he can walk at a speed 1m/ s. In the central square, he can walk only at a speed of $\upsilon\text{m/ s}(\upsilon<1)$. What is smallest value of v for which he can reach faster via a straight path through the sand than any path in the square outside the sand?

What is meant by limit of elasticity? Write Hooke's law of elasticity. Based on this establish the formula of Young's Modulus of elasticity and give the definition of Young's Modulus of elasticity.

A fully loaded Boeing aircraft has a mass of 3.3 × 10⁵kg. Its total wing area is 500m². It is in level flight with a speed of 960km/h.

Estimate the pressure difference between the lower and upper surfaces of the wings.
Estimate the fractional increase in the speed of the air on the upper surface of the wing relative to the lower surface. [Density of air = 1.2kg m^-3].

A metre scale AB is held vertically with its one end A on the floor and is then allowed to fall. Find the speed of the other end B when it strikes the floor, assuming that the end on the floor does not slip.

Explain Aristotle's Fallacy and Galileo's refutation.

Calculate the angular momentum and rotational kinetic energy of earth about its own axis. How long could this amount of energy supply one kilowatt power to each of the 3.5 × 10⁹ persons on earth? (Mass of earth = 6.0 × 1024kg and radius = 6.4 × 10²⁴km).