Advertisements
Advertisements
Question
Define angular S.H.M. and obtain its differential equation.
Solution
Angular S.H.M. is defined as the oscillatory motion of a body in which the torque for angular acceleration is directly proportional to the angular displacement and its direction is opposite to that of angular displacement.
- Consider a metallic disc hanging from rigid support, when twisted, it performs an oscillatory motion for which the restoring torque acting upon it, for angular displacement θ is,
τ ∝ -θ
∴ τ = -cθ ….(1) - The constant of proportionality (c) is the restoring torque per unit angular displacement.
- If I is the moment of inertia of the disc, the torque acting on the disc is given by,
τ = Iα .....(2)
Where α is the angular acceleration. - From equations (1) and (2),
Iα = -cθ
∴ `"I"("d"^2θ)/"dt"^2 + "c"θ = 0` .........`(∵ α = ("d"^2θ)/("dt"^2))`
This is the differential equation for angular S.H.M.
RELATED QUESTIONS
Prove that under certain conditions a magnet vibrating in a uniform magnetic field performs angular S.H.M.
An α-particle of energy 10 eV is moving in a circular path in uniform magnetic field. The energy of proton moving in the same path and same magnetic field will be [mass of α-particle = 4 times mass of proton]
When a particle carrying a charge of 100 µC moves at an angle of 30° to a uniform magnetic field of induction 6 x 10-5 Wb/m2 with a speed of 2 x 105 m/s. The force acting on the particle is ______
An `alpha` - particle of energy 5 MeV is scattered through 180° by gold nucleus. The distance of closest approach is of the order of ______.
A conductor in the fom1 of a right angle `\triangle` ABC with AB = 3 cm and BC = 4 cm carries a current of 15 A. There is a uniform magnetic field of 8 T perpendicular to the plane of the conductor. The force on the conductor will be ______.
Write the differential equation for angular S.H.M.
An α -particle of energy 5 Me V is scattered through 180° by a fixed uranium nucleus. The distance of closest approach is of the order of ______.
A bar magnet of mass 120 g in the form of a rectangular parallelepiped, has dimensions l = 40 mm, b = 10 mm and h = 80 mm, with its dimension ‘h’ vertical, the magnet performs angular oscillations in the plane of the magnetic field with period π seconds. If the magnetic moment is 3.4 Am2, determine the influencing magnetic field.
A particle executes S.H.M. of amplitude 3 cm. Its acceleration at extreme position is 27 cm/s2 Calculate its angular velocity.
