SHM and its equation, phase
- 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 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 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