1) Find the tension in the pendulum at the extreme position if amplitude is q .
2) A sphere of mass 0.2 kg is attached to an inextensible string of length of 0.5 m whose upper end is fixed to the celling . the sphere is made to describe a horizontal circle of radius 0.3m . the speed of the sphere will be ?
3) Two wires ac and bc are tied at c of small sphere of mass 5kg , which revolves at a constant speed v in the horizontal circle of radius 1.6 m . the minimum value of v is ?
4) A man is supported on a frictionless horizontal surface . it is attached to a spring and rotates about a fixed centre at an angular velocity 20. The tension in the string is f, of the length of string and angular velocity are doubled , the tension in string is now?
5) A car of mass 1000kg moves on a circular track of radius 20m . if the coefficient of friction is 0.64 , then the maximum velocity with which the car can moves is?
6) Moment of inertia of a thin rod of mass m and length l about an axis passing through its centre is ml/m its moment of inertia about a parallel axis at a distance of l/4 from this axis is given by ?
7) A solid cylinder of mass 20 kg has length 1m and radius 0.2m . then its moment of inertia in kg m2 about its geometrical axis is?
Here,
Mass, m = 0.2 kg
Length, L = 0.5 m
Radius of horizontal circle, r = 0.3 m
Let ‘v be the speed of the sphere. ‘θ’ be the angle made by the string with the vertical.
From the diagram,
OQ2 = OA2 + QA2
=> L2 = OA2 + r2
=> OA2 = L2 – r2
=> OA = (0.52 – 0.32)1/2
=> OA = (0.25 – 0.09)1/2
=> OA = 0.4 m
So, cosθ = OA/OQ
=> cosθ = 0.4/0.5
=> cosθ = 0.8
T cosθ = mg
=> T(0.8) = (0.2)(9.8)
=> T = 2.45 N
And,
T sinθ = mv2/r
=> T(r/L) = mv2/r
=> (2.45)(0.3/0.5) = (0.2)(v2)/(0.3)
=> v = 1.48 m/s
This is the speed of the sphere.