Advertisements
Advertisements
Question
What is a conical pendulum? Obtain an expression for its time period
Advertisements
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
A thin walled hollow cylinder is rolling down an incline, without slipping. At any instant, without slipping. At any instant, the ratio "Rotational K.E.: Translational K.E.: Total K.E." is ______.
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?
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 `sqrt 10` 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?
During ice ballet, while in the outer rounds, why do the dancers outstretch their arms and legs.
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 body weighing 0.5 kg tied to a string is projected with a velocity of 10 m/s. The body starts whirling in a vertical circle. If the radius of the circle is 0.8 m, find the tension in the string when the body is at the top of the circle.
Derive an expression for the kinetic energy of a rotating body with uniform angular velocity.
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, ______
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 is the relation between torque and angular momentum?
What are the rotational equivalents for the physical quantities, (i) mass and (ii) force?
Discuss conservation of angular momentum with example.
A uniform metallic rod rotates about its perpendicular bisector with constant angular speed. If it is heated uniformly to raise its temperature to a certain value, its speed of rotation ______.
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 ______.
A ring and a disc of different masses are rotating with the same kinetic energy. If we apply a retarding torque τ on the ring, it stops after completing n revolution in all. If the same torque is applied to the disc, how many revolutions would it complete in all before stopping?
What is the difference between rotation and revolution?
