Give two examples of periodic motion which are not oscillatory? | Physics

OSCILLATIONS

Very Short Question Answers

Q1. Give two examples of periodic motion which are not oscillatory?
Ans.

Revolution of planets around the sun.
The revolution of the electrons around the nucleus

Q2. The displacement in S.H.M. is given by y = a sin (20t +4). What is the displacement whenit is increased by 2?/? ?
Ans.

In S.H.M, the particle comes to same position, for every time period 𝑇 = 2π/ω. So
displacement of the particle remains same even after time is increased by 2π/ω

Q3. A girl is swinging seated in a swing. What is the effect on the frequency of oscillation if the stands?
Ans.

When the girl stands up, her centre of mass moves up and length of swing decreases. Therefore, frequency of oscillation increases

$\nu \propto \frac{1}{\sqrt{L}}$

Q4. 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?
Ans.

When water begins to drain out of sphere, its C.M shifts downward. The length of a pendulum and Time period also increases. When the entire water is drained out of the sphere, its C.M shift to entre of sphere and the time period (T) attains its initial value.

Q5. The bob of a simple pendulum is made of wood. What will be the effect on the time period if the wooden bob is replaced by an identical bob of aluminum?
Ans.

Time period remains same because time period does not depend up on the nature of the material of the bob.

Q6. Will a pendulum clock gain or lose time when taken to the top of a mountain?
Ans.

$Time\: Period\: \nu \propto \sqrt \frac {L}{g}$

on the top of mountain, value of g is less therefore period of oscillation increases and clock loses time.

Q7. A pendulum clock gives correct time at the equator. Will it gain or lose time if it is taken to the poles? If so, why?
Ans.

$Time\:\: Period \:\:T \propto \frac{1}{\sqrt{g}}$

on the top of mountain, value of g is less therefore period of oscillation increases and clock loses time.

Q8. What fraction of the total energy is K.E when the displacement is one half of a amplitude of particle executing S.H.M.?
Ans.

$https://latex.codecogs.com/png.image?\inline \large \dpi{110}\\Total\: Energy \: E= \frac{1}{2}kA^{2}, Displacement \:\: Y=\frac{A}{2}\\\:\: Kinetic\:Energy \:\:K.E=\frac{1}{2}k(A^{2}-Y^{2})\\\therefore \:\: K.E=\frac{1}{2}k(A^{2}-({\frac{A}{2}})^{2})=\frac{3}{4}(\frac{1}{2}kA^{2})\\K.E=\frac{3}{4}E$

Q9. What happens to the energy of a simple harmonic oscillator if its amplitude is doubled?
Ans.

$\\ A_{1}=A,\:\: A_{2}=\frac{A}{2}\\\frac{E_2}{E_1}=(\frac{A_1}{A_2})^2 \\ \frac{E_2}{E_1}=(\frac{A}{A/2})^2\\ \frac{E_2}{E_1}=4\\E_{2}=4E_{1} \\\therefore The energy is increased by 4 times.$

Q10. Can the pendulum clocks be used in an artificial satellite?
Ans.

No, in artificial satellite the acceleration due to gravity is zero (𝒈 = 𝒐). So simple pendulum cannot oscillate when it is used in artificial satellite.

Short Question Answers

Q1. Define simple harmonic motion. Give two examples?

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Q2. Present graphically the variations of displacement, v locity and acceleration with time for a particle in S.H.M?

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Q3. What is phase? Discuss the phase relations between displacement, velocity and acceleration in simple harmonic motion?

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Q4. Obtain an equation for frequency of oscillation of a spring of force constant K to which a mass m is attached?

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Q5. Derive the expression for the kinetic energy and potential energy of simple harmonic oscillator?

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Q6. How does the energy of a simple pendul m vary as it moves from one extreme position to the other during its oscillation?

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Q7. Derive the expression for displacement, velocity and acceleration of a particle executes SHM?

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Essay Question Answers

Q1. Define Simple Harmonic Motion?Show that the projection of uniform circular motion on any diameter is simple harmonic?

Ans. Simple Harmonic Motion: The 'to and fro motion' of a particle about a fixed point is said to be 'simple harmonic motion'. when

The acceleration is directly proportional to its displacement, in opposite.
The acceleration is always towards the fixed point

Proof: Consider a particle $P$ moving on the circumference of a circle of radius $A$ with uniform angular velocity $\omega$. As particle $P$ moves on the circumference of the circle, projection $N$ moves to and fro on the diameter $YY'$ about the $0$

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Displacement : Form the $\triangle OPN$ $$\begin{aligned}
&&\sin\theta={\frac{ON}{OP}}={\frac{y}{A}} \\
&&\mathbf{y}=\mathbf{A}\sin\theta \\
&&\mathrm{y}=\mathrm{A}\sin\omega t \end{aligned}$$

Velocity:$$\begin{aligned}
&&V={\frac{dy}{dt}} \\
&&V={\frac{d}{dt}}(A\sin\omega t) \\
&&V=\:\mathrm{A}\omega\:\mathrm{Cos}\:\omega t
\end{aligned}$$Acceleration:$$\begin{gathered}
a={\frac{dv}{dt}} \\
a={\frac{d}{dt}}(\mathrm{A}\omega\:\mathrm{Cos}\:\omega t) \\
a=-A\omega^{2}\sin\omega t \\
a=-\omega^{2}y \\
a\propto-y \\
\end{gathered}$$

Hence, the motion of $'N'$ is simple harmonic motion.The projection of uniform circular motion on any diameter is simple harmonic.

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

Ans. Derivation:A bob of mass $'m'$ oscillating in aplane about thevertical line through the support. There are only two forces acting on the bob.

i)Tension $T$ due to the string, Vertical force due to gravity $mg$

ii) The force $mg$ can be resolved into two components $mg \:cos\theta$ , $mg \sin\theta$

iII) $mg \cos\theta$ balance with the tension $ (T)$

iv)$mg\sin⁡θ$ provides restoring force.

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$$\begin{aligned}
F=-\:mg\sin\theta \\
ma=-\:mg\sin\theta \\
a=-\:g\sin\theta \\
a=-g\theta \\
a=-\:g\left({\frac{x}{l}}\right) \\
a=-\left({\frac{g}{l}}\right)x \\
a\propto-x
\end{aligned}$$Where $g$ and $l$ are constants. Hence, the motion of the simple pendulum is simple harmonic.

Equation for Time Period: from the above equations angular velocity $\omega=\sqrt{\frac{g}{\iota}}$

$$\begin{aligned}&\therefore Time\:period\:T=\frac{2\pi}{\omega}\\&T=2\pi\sqrt{\frac{l}{g}}\\\end{aligned}$$

Seconds Pendulum: A simple pendulum whose time period is two seconds is called seconds pendulum

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

Ans. Consider a particle $P$ moving on the circumference of a circle of radius $A$ with uniform angular velocity $\omega $ .At a time $t$ The displacement of the particle is $\mathbf{y}=A sin(\omega t)$.Hence, velocity is $V =A\omega cos(\omega t)$.

kinetic energy $K=\frac{1}{2}mv^{2}$ $$\begin{array}{}{K=\frac{1}{2}m(\mathrm{A}\omega\:\mathrm{Cos}\:\omega t)^{2}}\\{K=\frac{1}{2}mA^{2}\omega^{2}cos^{2}\theta } \end{array}$$ Work done for a small displacement $ dy $ is given by $dw=Fdy$

The work done is $W=\frac{1}{2}m\omega^{2}y^{2}$,in simple harmonic motion while moving towards extreme positionsfrom mean position,it will in the form of potential energy.$$\begin{array}{}{\therefore U=\frac{1}{2}m\omega^{2}y^{2}}\\{U=\frac{1}{2}mA^{2}\omega^{2}sin^{2}\theta} \end{array}$$Therefore the total energy of the particle is$$\begin{array}{} {E=K+U}\\{=\frac{1}{2}mA^{2}\omega^{2}cos^{2}\theta+\frac{1}{2}mA^{2}\omega^{2}cos^{2}\theta}\\{E=\frac{1}{2}mA^2\omega^2}\end{array}$$The above equation is independent of time.Hence,The total energy of a particle in simple harmonic motion is constant at any point on it's path.

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Problems Question Answers

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?

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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?

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

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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?

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Q5. A mass of 2kg attached to a spring of force constant What is the time taken?

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Q6. A simple pendulum in a stationary life has time period T. What would be the effect on the time period when the lift i) Move Up with uniform velocity ii) Moves Do n with uniform velocity iii) Moves Up with uniform acceleration a iv) Moves Down with uniform acceleration ‘a’ v) Begins To fall freely under gravity?

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Q7. A particle executing SHM has amplitude of 4cm nd its acceleration at a distance of 1cm from the mean position is 2cm/s² . What will be its velocity when it is at a distance of 2cm from its mean position?

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Q8. A simple harmonic oscillator has a time period of 2s.What will be the change in the phase 0.25 s after leaving the mean position?

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

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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?

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Q11. Calculate the change in the length of a simple pendulum of length 1m, when its period of oscillation changes from 2s to 1.5s?

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Q12. A freely falling body takes 2 seconds to reach the ground on a planet, when it is dropped from a height of 8m. If the period of a simple pendulum is ? seconds on the planet, calculate the length of the pendulum?

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Q13. The period of a simple pendulum is found to increase by 50% when the length of the pendulum is increased by 0.4m. Calculate the initial length and the initial period of oscillation at a place where g=9.8m/s²

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Q14. A clock regulated by a second’s pendulum keeps correct time. During summer the length of the pendulum increases to 1.02m. How much will the clock gain or lose in one day?

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

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