Oscillations
Oscillations
- The displacement of a particle in SHM is \(x=5\sin\left(\pi t+\frac{\pi}{3}\right)\) cm. Its maximum speed is Answer: B. 5π cm/s v_max=Aω=5×π=5π cm/s. 1
- A mass attached to a spring oscillates with period \(T\). The spring is cut into two equal halves and the same mass is attached to one half. The new period is Answer: D. T/sqrt2 Half spring has k'=2k . T∝1/sqrt k⇒ T'=T/sqrt2 . 1
- A wooden block of mass m is kept on a piston that Given : A = 0.25m, g = π² = 10 m s² During vertical oscillations, the acceleration is maximum at the turning points at the top and bottom. T... 1
- Find the number of oscillations performed per minute by a magnet is vibrating in the plane of a uniform field Given: B =1.6 x 10⁻⁵ T, I = 3 x 10⁻⁶ Kg/m², m =3 A-m² The period of oscillation, T = 2π √(I/μB) ∴ The frequency of oscillation is η = 1/2π√(... 1
- A 20 cm wide thin circular disc of mass 200 g is suspended to a rigid support from a thin metallic string Given: R = 10cm = 0.1 m, M = 0.2 kg, θ_m=60⁰ = p/3 rad, T=1 s, π³ = 31 The MI of the disc about the rotation axis (perpendicular through its... 1
- Two parallel S.H.M.s represented by x1 = 5sin (4πt + π/3) cm and x2 = 3sin (4πt + π/4) cm are superposed Given : x₁ = 5sin (4πt + π/3) = A₁ sin(ωt + α), x₂ = 3sin (4πt + π/4) = A₂ sin(ωt + β), ∴ A₁ =5 cm, A₂ =3 cm, α = π/3 rad, β = π/4 rad (i) R... 1
- The displacement of an oscillating particle is given by x= a sin ωt + b cos ωt where a, b and ω are constant x= a sin ωt + b cos ωt Let a = A cos ∅ and b = A sin ∅, so that A² = a^(2 )+ b² and tan ∅ = b/a x=A cos ∅ sin ωt + A sin ∅ cos ωt x = A sin... 1
- The period of oscillation of a body of mass m1 suspended from a light spring is T. For mass m₁, period is T ∴ T = 2π √(m₁/k) For mass m₂, period is 2T ∴ 2T = 2π √(m₁+m₂/k) ∴ 2T/T = 2 = √(m₁+m₂/m₁) ∴ m₁+m₂/m₁ = 4 ∴ m₂/m₁ = 3... 1
- A particle performing linear S.H.M. of period 2p seconds Given : T = 2π s, v = b√(3) m/s at x=b ω = 2π/T = 2π/2π =1 rad/s v = ω √(A²-x²) ∴ at x = b, b√(3) = (1) √(A²-x²) ∴ 3b² = A² – b^(2 ), ∴ A =... 1
- At what distance from the mean position is the kinetic energy of a particle performing S.H.M. of amplitude 8 cm Given : A=8 cm, KE = 3PE KE = 1/2 k(A² —x²) and PE = 1/2 kx² Given, KE = 3PE. ∴ 1/2 k(A² —x²) = 3( 1/2 kx²) ∴ A² —x^(2 )= 3x^(2 ), ∴ 4x² = A... 1
- Find the change in length of a second’s pendulum, if the acceleration due to gravity at the place changes from 9.75 Given : g₁ =9.75 m/s², g₂ = 9.8 m/s² Length of a seconds pendulum, L = g/π² ∴ L₁ = g₁/π² = 9.75/9.872 = 0.9876 m and ∴ L₂ = g₂/π^(2 )= 9.8/9... 1
- A simple pendulum of length 100 cm performs S.H.M. Find the restoring force acting on its bob of mass Given : L = 100cm, m = 50 g = 5 x 10⁻² kg, x = 3 cm, g = 9.8 m/s² Restoring force, F = mg sin θ = mgθ = (5 x 10⁻²)(9.8)(3/100) = 1.48 x 10⁻²... 1
- A simple pendulum performs S.H.M of period 4 seconds. How much time after crossing the mean position Given : T = 4 s, x = A/3 The displacement of a particle starting into SHM from the mean position is x =A sin ωt = A sin(2π/T)t ∴ A sin(2π/T)... 1
- The total energy of a body of mass 2 kg performing S.H.M. is 40 J. Find its speed while crossing the centre of the path. Given : m = 2 kg, E = 40 J The speed of the body while crossing the centre of the path (mean position) is v_max and the total energy is enti... 1
- Potential energy of a particle performing linear S.H.M is 0.1 π2 x2 joule. If mass of the particle is 20 g, Given : PE = 0.1π²x^(2 )J, m = 20 g = 2 X 10⁻² kg PE = ½ mw²x² = ½m(4π²η²)x^(2 )= 0.1π²x² ∴ 2mη² = 0.1 ∴ η² = 0.1/[2(2 X 10⁻²)]= 2.5 ∴ The f... 1
- A needle of a sewing machine moves along a path of amplitude 4 cm with frequency 5 Hz. Given : A=4 cm = 4 x 10⁻²m, η = 5Hz, t= 1/30 s ω = 2πη = 2π x 5 = 10π rad/s Therefore, the magnitude of the acceleration, |a| = ω²x = ω²A si... 1
- In SI units, the differential equation of an S.H.M. is \(\frac{d^2x}{dt^2}=-36x\) . Find its frequency and period. Given : d²x/dt²=-36x Comparing this equation with the general equation, We get, ω^(2 )= 36, ∴ ω = 6 rad/s ω = 2πη The frequency, η = ω/2π =... 1
- At what distance from the mean position is the speed of a particle performing S.H.M. half its maximum speed Given : v = 1/2v_max , 2A=10 cm A=5 cm v = ω√(A²-x²) and v_max = ωA Since v = 1/2v_max ω√(A²-x²)=ωA/2 ∴ A²-x²=A²/4 ∴ x²=A²-A²/4 = 3A²/4 ∴ x=... 1
- Prove that under certain conditions a magnet vibrating in uniform magnetic field performs angular S.H.M. Consider a bar magnet of magnetic moment μ, suspended horizontally by a light twistless fibre in a region where the horizontal component of... 1
- State the laws of simple pendulum. The period of a simple pendulum at a given place is T = 2π√(L/g) where L is the length of the simple pendulum and g is the acceleration due... 2
- Obtain the expression for the period of a simple pendulum performing S.H.M. An ideal simple pendulum is defined as a heavy point mass suspended from a rigid support by a weightless, inextensible and twistless string,... 1
- Using differential equation of linear S.H.M, obtain the expression for (a) velocity in S.H.M., (b) acceleration in S.H.M The general expression for the displacement of a particle in SHM at time t is x = A sin(ωt + α) ….. (1) where A and ω is a constant in a par... 2
- Define linear simple harmonic motion. Linear simple harmonic motion (SHM) is defined as the linear periodic motion of a body, in which the force (or acceleration) is always direc... 3
- The graph shows variation of displacement of a particle performing S.H.M. with time t. (B) The force is maximum at time 3T/4. 1