Advertisements
Advertisements
Question
What is a conical pendulum? Obtain an expression for its time period
Solution
A tiny mass (assumed to be a point object and called a bob) connected to a long, flexible, massless, inextensible string, and suspended to rigid support revolves in such a way that the string moves along the surface of a right circular cone of the vertical axis and the point object performs a uniform horizontal circular motion. Such a system is called a conical pendulum.
Expression for its time period:
- Consider the vertical section of a conical pendulum having bob (point mass) of mass m and string of length ‘L’.
Here, θ is the angle made by the string with the vertical, at any position (semi-vertical angle of the cone) - In a given position B, the forces acting on the bob are
a. its weight ‘mg’ directed vertically downwards
b. the force ‘T0’ due to the tension in the string, directed along the string, towards the support A.
In an inertial frame - As the motion of the bob is a horizontal circular motion, the resultant force must be horizontal and directed towards the centre C of the circular motion.
For this, tension (T0) in the string is resolved into
a. T0 cos θ: vertical component
b. T0 sin θ: horizontal component - The vertical component (T0 cos θ) balances the weight ‘mg’.
∴ mg = T0 cosθ …..............(1)
The horizontal component T0 sinθ then becomes the resultant force which is centripetal.
mrω2 = T0 sinθ …..............(2)
Dividing equation (2) by equation (1),
ω2 = `(gsinθ)/(rcosθ)` …..............(3) - From the figure,
sinθ = `r/L`
∴ r = L sin θ …..............(4)
From equation (3) and (4),
ω2 = `(gsinθ)/(L.sinθ.cosθ)`
ω = `sqrt(g/(Lcosθ)` - If T is the period of revolution of the bob, then
ω = `(2pi)/T = sqrt(g/(Lcosθ)`
∴ Period, T = `2pisqrt((Lcosθ)/g)`
RELATED QUESTIONS
Answer in brief:
Why are curved roads banked?
Do we need a banked road for a two-wheeler? Explain.
On what factors does the frequency of a conical pendulum depend? Is it independent of some factors?
While driving along an unbanked circular road, a two-wheeler rider has to lean with the vertical. Why is it so? With what angle the rider has to lean? Derive the relevant expression. Why such a leaning is not necessary for a four wheeler?
Somehow, an ant is stuck to the rim of a bicycle wheel of diameter 1 m. While the bicycle is on a central stand, the wheel is set into rotation and it attains the frequency of 2 rev/s in 10 seconds, with uniform angular acceleration. Calculate:
- The number of revolutions completed by the ant in these 10 seconds.
- Time is taken by it for first complete revolution and the last complete revolution.
The coefficient of static friction between a coin and a gramophone disc is 0.5. Radius of the disc is 8 cm. Initially the center of the coin is 2 cm away from the center of the disc. At what minimum frequency will it start slipping from there? By what factor will the answer change if the coin is almost at the rim? (use g = π2m/s2)
Answer in Brief:
A flywheel used to prepare earthenware pots is set into rotation at 100 rpm. It is in the form of a disc of mass 10 kg and a radius 0.4 m. A lump of clay (to be taken equivalent to a particle) of mass 1.6 kg falls on it and adheres to it at a certain distance x from the center. Calculate x if the wheel now rotates at 80 rpm.
Starting from rest, an object rolls down along an incline that rises by 3 in every 5 (along with it). The object gains a speed of `sqrt10` m/s as it travels a distance of `5/3` m along the incline. What can be the possible shape/s of the object?
Does the angle of banking depend on the mass of the vehicle?
A hollow sphere has a radius of 6.4 m. what is the minimum velocity required by a motorcyclist at the bottom to complete the circle.
A bucket containing water is tied to one end of a rope 5 m long and it is rotated in a vertical circle about the other end. Find the number of rotations per minute in order that the water in the bucket may not spill.
A railway track goes around a curve having a radius of curvature of 1 km. The distance between the rails is 1 m. Find the elevation of the outer rail above the inner rail so that there is no side pressure against the rails when a train goes around the curve at 36 km/hr.
Obtain an expression for maximum safety speed with which a vehicle can be safely driven along a curved banked road.
A rigid body rotates with an angular momentum L. If its kinetic energy is halved, the angular momentum becomes, ______
A particle undergoes uniform circular motion. The angular momentum of the particle remains conserved about, ______
When a mass is rotating in a plane about a fixed point, its angular momentum is directed along, ______
Give any two examples of torque in day-to-day life.
What are the rotational equivalents for the physical quantities, (i) mass and (ii) force?
Discuss conservation of angular momentum with example.
A flywheel rotates with uniform angular acceleration. If its angular velocity increases from `20pi` rad/s to `40pi` rad/s in 10 seconds. Find the number of rotations in that period.
A wheel of radius 2 cm is at rest on the horizontal surface. A point P on the circumference of the wheel is in contact with the horizontal surface. When the wheel rolls without slipping on the surface, the displacement of point P after half rotation of wheel is ______.
What is the difference between rotation and revolution?
