Oscillations and Waves
- Two identical point masses P and Q, suspended from two separate massless springs of spring constants k_1 and k_2, respectively, oscillate vertically. If their maximum speeds are the same, the ratio (A_Q/A_P) of the amplitude A_Q of mass Q to the ampl… Answer: 4. √(k₁/k₂) Key idea: the maximum speed in SHM is v_max=Aω=A√(k/m) . Masses are identical and v_max is the same, so A_P√(k₁)=A_Q√(k₂... 1
- A pipe open at both ends has a fundamental frequency f in air. The pipe is now dipped vertically in a water drum to half of its length. The fundamental frequency of the air column is now equal to : Answer: 2. f Key idea: an open pipe of length L has fundamental f=v/2L ; a pipe closed at one end of length l has fundamental v/4l . Dipping... 1
- In an oscillating spring mass system, a spring is connected to a box filled with sand. As the box oscillates, sand leaks slowly out of the box vertically so that the average frequency ω(t) and average amplitude A(t) of the system change with time t.… Answer: 2. Graph pair: ω(t) rises and saturates; A(t) decays Key idea: the frequency of a spring-mass system is ω=√(k/m) , and sand leaking... 1
- If the mass of the bob in a simple pendulum is increased to thrice its original mass and its length is made half its original length, then the new time period of oscillation is x/2 times its original time period. Then the value of x is: Answer: 3. sqrt2 Key idea: T=2π√(L/g) does not depend on the mass of the bob. Only the length changes: T'=2π√(L/2/g)=T/sqrt2 . Setting T/sqr... 1
- If x=5≤ft(π t+π/3)m represents the motion of a particle executing simple harmonic motion, the amplitude and time period of motion, respectively, are : Answer: 3. 5 m, 2 s Key idea: Compare with the standard SHM form x=Asin(ω t+φ) . Here A=5 m and ω=π rad/s. Time period T=2π/ω=2π/π=2 s. Ampl... 1
- The x-t graph of a particle performing simple harmonic motion is shown in the figure. The acceleration of the particle at t=2 s is : Answer: 1. -π²/16 mathrmm s⁻² From the graph, the amplitude is A=1 m and the particle is at the maximum at t=2 s, with a zero crossing at 4... 1
- The ratio of frequencies of fundamental harmonic produced by an open pipe to that of closed pipe having the same length is : Answer: 3. 2 : 1 Fundamental frequency of an open pipe: f_o=v/2L . For a closed pipe: f_c=v/4L . f_o/f_c=4L/2L=2:1 . The answer is 2 : 1. 1
- Two pendulums of length 121 cm and 100 cm start vibrating in phase. At some instant, the two are at their mean position in the same phase. The minimum number of vibrations of the shorter pendulum after which the two are again in phase at the mean pos… Answer: 2. 11 Time period T∝sqrt l , so T₁₂₁:T₁₀₀=11:10 . They are next in phase when the shorter pendulum has made one more vibration than... 1
- If the initial tension on a stretched string is doubled, then the ratio of the initial and final speeds of a transverse wave along the string is : Answer: 4. 1 : √(2) Speed of a transverse wave on a string is v=√(T/μ) , so v∝sqrt T . v_i/v_f=√(T/2T)=1/sqrt2 , i.e. 1:sqrt2 . Option (4) i... 1
- A body is executing simple harmonic motion with frequency 'n', the frequency of its potential energy is : Answer: 2. 2n Key idea: potential energy depends on x² , so it repeats twice per oscillation. x = Asinω t ⇒ U=1/2 kx²=1/2 kA²sin²ω t=1/4 kA²... 1
- A spring is stretched by 5 cm by a force 10 N. The time period of the oscillations when a mass of 2 kg is suspended by it is : Answer: 4. 0.628 s Key idea: find k from the stretch, then use T=2π√(m/k) . k=F/x=10/0.05=200\ N/m T=2π√(2/200)=2π×0.1=0.628\ mathrm s . Opt... 1
- In a guitar, two strings A and B made of same material are slightly out of tune and produce beats of frequency 6 Hz. When tension in B is slightly decreased, the beat frequency increases to 7 Hz. If the frequency of A is 530 Hz, the original frequenc… Answer: 2. 524 Hz Frequency of a string f∝√(T) . Decreasing tension of B lowers its frequency. Beat frequency was 6 Hz and rose to 7 Hz afte... 1
- The phase difference between displacement and acceleration of a particle in a simple harmonic motion is : Answer: 1. π rad For SHM, x=Asinω t , and acceleration a=-ω²x . The negative sign means acceleration is always opposite in direction to disp... 1
- The radius of circle, the period of revolution, initial position and sense of revolution are indicated in the fig.y-projection of the radius vector of rotating particle P is : Answer: 4. y(t) = 3cos(π t/2) , where y in m From the figure, A=3 m, T=4 s so ω=2π/T=π/2 rad/s. At t=0 the particle is at the top of the cir... 1
- The displacement of a particle executing simple harmonic motion is given byy = A_0 + Aω t + Bω t.Then the amplitude of its oscillation is given by : Answer: 2. √(A²+B²) The constant A₀ only shifts the mean position. The oscillating part is Asinω t+Bcosω t . Write it as Csin(ω t+φ) with Cc... 1
- Average velocity of a particle executing SHM in one complete vibration is : Answer: 4. zero Average velocity =displacement/time . In one complete vibration the particle returns to its starting point, so the displacem... 1
- A pendulum is hung from the roof of a sufficiently high building and is moving freely to and fro like a simple harmonic oscillator. The acceleration of the bob of the pendulum is 20 m/s^2 at a distance of 5 m from the mean position. The time period o… Answer: 3. π s For SHM, a=ω²x . ω²=20/5=4⇒ω=2\ mathrmrad s⁻¹ . T=2π/ω=π\ s 1
- A particle executes linear simple harmonic motion with an amplitude of 3 cm. When the particle is at 2 cm from the mean position, the magnitude of its velocity is equal to that of its acceleration. Then its time period in seconds is : Answer: 2. 4π/sqrt5 Key idea: in SHM v=ω√(A²-x²) and a=ω²x . At x=2 cm, A=3 cm: ω√(9-4)=ω²×2⇒ω=sqrt5/2 s ⁻¹ . T=2π/ω=4π/sqrt5 s. 1
- The fundamental frequency in an open organ pipe is equal to the third harmonic of a closed organ pipe. If the length of the closed organ pipe is 20 cm, the length of the open organ pipe is Answer: 1. 13.2 cm Closed pipe: odd harmonics only, f_n=nv/4L . Third harmonic: f=3v/4× 20 . Open pipe fundamental: f₁=v/2L' . Equating: v/2... 1
- A spring of force constant k is cut into lengths of ratio 1 : 2 : 3. They are connected in series and the new force constant is k'. Then they are connected in parallel and force constant is k''. Then k' : k'' is : Answer: 2. 1 : 11 Key idea: spring constant is inversely proportional to length. Pieces have lengths L/6,L/3,L/2 , so constants 6k,3k,2k . S... 1
- The two nearest harmonics of a tube closed at one end and open at other end are 220 Hz and 260 Hz. What is the fundamental frequency of the system ? Answer: 1. 20 Hz Key idea: a tube closed at one end has only odd harmonics: f₀,3f₀,5f₀,dots . Consecutive harmonics differ by 2f₀ . 260-220=... 1
- Two cars moving in opposite directions approach each other with speed of 22 m/s and 16.5 m/s respectively. The driver of the first car blows a horn having a frequency 400 Hz. The frequency heard by the driver of the second car is [velocity of sound 3… Answer: 3. 448 Hz Key idea: Doppler effect with source and observer approaching: f'=fv+v_o/v-v_s . f'=400×340+16.5/340-22=400×356.5/318≈448... 1
- Three sound waves of equal amplitudes have frequencies (n-1), n, (n+1). They superimpose to give beats. The number of beats produced per second will be Answer: 3. 2 Add the three waves of equal amplitude and see how the intensity repeats. y=A[cos2π(n-1)t+cos2π nt+cos2π(n+1)t]=Acos2π nt (1+2c... 1
- A body of mass m is attached to the lower end of a spring whose upper end is fixed. The spring has negligible mass. When the mass m is slightly pulled down and released, it oscillates with a time period of 3 s. When the mass m is increased by 1 kg, t… Answer: 3. 9/16 Time period of a spring-mass system is T=2π√(m/k) , so T²∝ m . T₁²/T₂²=m/m+1⇒9/25=m/m+1 9m+9=25m , so 16m=9 and m=9/16 kg. O... 1