Q1. Give two examples of periodic motion.

Q2. A girl is swinging while seated on a swing. What happens to the frequency of oscillation when she stands up?

Q3. The bob of a simple pendulum is a hollow sphere, filled with water. How will the period of oscillation change if the water begins to drain out of the hollow sphere?

Q4. Will the pendulum lose or gain time when taken to the top of a mountain?

Q5. What happens to the energy of a simple harmonic oscillator if its amplitude is doubled?

Q6. Can a simple pendulum be used in artificial satellite?

Q7. Can a simple pendulum oscillate at the center of the earth? Why?

Q8. What happens to the time period of a simple pendulum, if its length is doubled?

Q9. A block of mass \(m\) when hung on spiral spring stretches by \(20\,\mathrm{cm}\). What is its period of oscillation when pulled down and released?

Q10. Write the expression for the total energy of a simple harmonic oscillator.

Q11. What is force constant? Write its units in S.I. system.

Q1. Define simple harmonic motion. Give two examples.

Q2. Obtain an expression for frequency of oscillation of a spring of force constant \(k\) to which a mass \(m\) is attached.

Q3. Derive an expression for kinetic energy of simple harmonic oscillator.

Q4. Derive an expression for potential energy of simple harmonic oscillator.

Q5. Derive expressions for displacement, velocity and acceleration of a particle executing S.H.M.

Q6. How does the energy of a simple pendulum vary as it moves from one extreme position to other extreme position?

Q1. Define simple harmonic motion. Show that the motion of projection of a particle performing uniform circular motion on any diameter is simple harmonic.

Q2. Show that the motion of simple pendulum is simple harmonic and hence derive an equation for its time period. What is seconds pendulum?

Q3. Derive the equation for kinetic energy and potential energy of simple harmonic oscillator and show that the total energy of a particle in S.H.M is constant at any point on its path.

Q4. Show that oscillations due to a loaded spring are simple harmonic and derive an equation for its time period.

Q1. The bob of a pendulum is made of a hollow brass sphere. What happens to the time period of the pendulum, if the bob is filled with water completely? Why?

Q2. Two identical springs of force constant \(k\) are joined one at the end of the other (in series). Find the effective force constant of the combination.

Q3. What are the physical quantities having maximum value at the mean position in SHM?

Q4. A particle executes SHM such that the maximum velocity during the oscillation is numerically equal to half the maximum acceleration. What is the time period?

Q5. A mass of \(2\,\mathrm{kg}\) attached to spring of force constant \(260\,\mathrm{N\,m^{-1}}\) makes 100 oscillations. What is the time taken?

Q6. A simple pendulum in a stationery lift has time period T. What would be the effect on the time period when the lift (i) moves up with uniform velocity (ii) moves down with uniform velocity (iii) moves up with uniform acceleration a (iv) moves down with uniform acceleration a (v) begins to fall freely under gravity?

Q7. A particle executing SHM has amplitude of \(4\,\mathrm{cm}\), and its acceleration at a distance of \(1\,\mathrm{cm}\) from the mean position is \(3\,\mathrm{cm\,s^{-2}}\). What will be its velocity when it is at a distance of \(2\,\mathrm{cm}\) from its mean position?

Q8. A simple harmonic oscillator has a time period of \(2\,\mathrm{s}\). What will be the change in the phase after \(2.5\) seconds of time leaving the mean position?

Q9. A body describes simple harmonic motion with an amplitude of \(5\,\mathrm{cm}\) and a period of \(0.2\,\mathrm{s}\). Find the acceleration and velocity of the body when the displacement is (a) \(5\,\mathrm{cm}\), (b) \(3\,\mathrm{cm}\), (c) \(0\,\mathrm{cm}\).

Q10. The mass and radius of a planet are double that of the earth. If the time period of a simple pendulum on the earth is \(T\), find the time period on the planet.

Q11. Calculate the change in the length of a simple pendulum of length \(1\,\mathrm{m}\), when its period of oscillation changes from \(2\,\mathrm{s}\) to \(1.5\,\mathrm{s}\).

Q12. A freely falling body takes \(2\,\mathrm{s}\) to reach the ground on a planet, when it is dropped from a height of \(8\,\mathrm{m}\). If the period of a simple pendulum is \(\pi\) seconds on the planet, calculate the length of the pendulum.

Q13. The period of a simple pendulum is found to increase by 50% when the length of the pendulum is increased by \(0.6\,\mathrm{m}\). Calculate the initial length and the initial period of oscillation at a place where \(g=9.8\,\mathrm{m\,s^{-2}}\).

Q14. A clock regulated by a second's pendulum keeps correct time. During summer the length of the pendulum increases to \(1.02\,\mathrm{m}\). How much will the clock gain or lose in one day?

Q15. The time period of a body suspended from a spring is \(T\). What will be the new time period if the spring is cut into two equal parts and the mass is suspended (i) from one part (ii) simultaneously from both the parts?

Q1. Which of the following functions of time represent (a) periodic and (b) non-periodic motion? Give the period for each case of periodic motion [\(\omega\) is any positive constant]. (i) \(\sin\omega t+\cos\omega t\) (ii) \(\sin\omega t+\cos 2\omega t+\sin 4\omega t\) (iii) \(e^{-t}\) (iv) \(\log(\omega t)\).

Q2. Which of the following functions of time represent (a) simple harmonic motion and (b) periodic but not simple harmonic? Give the period for each case. (a) \(\sin\omega t-\cos\omega t\) (b) \(\sin^2\omega t\).

Q3. Fig. below depicts two circular motions. The radius of the circle, the period of revolution, the initial position and the sense of revolution are indicated on the figures. Obtain the simple harmonic motions of the x-projection of the radius vector of the rotating particle P in each case.

Q4. A body oscillates with SHM according to the equation (in SI units), \(x=5\cos(2\pi t+\pi/4)\). At \(t=1.5\,\mathrm{s}\), calculate the (a) displacement, (b) speed and (c) acceleration of the body.

Q5. Two identical springs of spring constant \(k\) are attached to a block of mass \(m\) and to fixed supports as shown in Fig. 7.14. Show that when the mass is displaced from its equilibrium position on either side, it executes a simple harmonic motion. Find the period of oscillation.

Q6. A block whose mass is \(1\,\mathrm{kg}\) is fastened to a spring. The spring has a spring constant of \(50\,\mathrm{N\,m^{-1}}\). The block is pulled to a distance \(x=10\,\mathrm{cm}\) from its equilibrium position at \(x=0\) on a frictionless surface from rest at \(t=0\). Calculate the kinetic, potential and total energy of the block when it is \(5\,\mathrm{cm}\) away from the mean position.

Q7. A metal ring of mass \(5\,\mathrm{kg}\) is attached to a spring of spring constant \(500\,\mathrm{N\,m^{-1}}\). It slides without friction over a horizontal rod. The ring is displaced from its equilibrium position by \(10.0\,\mathrm{cm}\) and released. Calculate (a) the period of oscillation, (b) the maximum speed and (c) maximum acceleration of the ring.

Q8. On an average a human heart is found to beat 75 times in a minute. Calculate its frequency and period.