Advertisements
Advertisements
प्रश्न
Derive an expression for the kinetic energy of a rotating body with uniform angular velocity.
Advertisements
उत्तर
- Consider a rigid object rotating with a constant angular speed ω about an axis perpendicular to the plane of the paper.
A body of N particles - For theoretical simplification, let us consider the object to be consisting of N particles of masses m1, m2, …..mN at respective perpendicular distances r1, r2, …..rN from the axis of rotation.
- As the object rotates, all these particles perform UCM with the same angular speed ω, but with different linear speeds,
v1 = r1ω, v2 = r2ω,…., vN = rNω - Translational K.E. of the first particle is
(K.E.)1 = `1/2m_1v_1^2 = 1/2m_1r_1^2omega^2`
Similar will be the case of all the other particles. - The rotational K.E. of the object is the sum of individual translational kinetic energies.
Thus,
Rotational K.E. = `1/2m_1r_1^2omega^2 + 1/2m_2r_2^2omega^2..... + 1/2m_Nr_N^2omega^2`
∴ Rotational K.E. = `1/2(m_1r_1^2 + m_2r_2^2..... + m_Nr_N^2)omega^2` - But I = `Σ_{i = 1}^N m_ir_i^2 = m_1r_1^2 + m_2r_2^2...... + m_Nr_N^2`
∴ Rotational K.E. = `1/2"I"omega^2`
APPEARS IN
संबंधित प्रश्न
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?
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.
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 bend in a level road has a radius of 100m. find the maximum speed which a car turning this bend may have without skidding if the coefficient of friction between the tires and road is 0.8.
Derive an expression for maximum safety speed with which a vehicle should move along a curved horizontal road. State the significance of it.
What is a conical pendulum? Obtain an expression for its time period
A rigid body rotates with an angular momentum L. If its kinetic energy is halved, the angular momentum becomes, ______
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?
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?
