Balbharati solutions for Physics [English] 12 Standard HSC Maharashtra State Board chapter 1 - Rotational Dynamics [Latest edition]

When seen from below, the blades of a ceiling fan are seen to be revolving anticlockwise and their speed is decreasing. Select the correct statement about the directions of its angular velocity and angular acceleration.

A particle of mass 1 kg, tied to a 1.2 m long string is whirled to perform the vertical circular motion, under gravity. The minimum speed of a particle is 5 m/s. Consider the following statements.

P) Maximum speed must be `5sqrt5` m/s.

Q) Difference between maximum and minimum tensions along the string is 60 N.

Select the correct option.

Choose the correct option.

Select correct statement about the formula (expression) of moment of inertia (M.I.) in terms of mass M of the object and some of its distance parameter/s, such as R, L, etc.

In a certain unit, the radius of gyration of a uniform disc about its central and transverse axis is `sqrt2.5`. Its radius of gyration about a tangent in its plane (in the same unit) must be ______.

Choose the correct option.

Consider the following cases:

(P) A planet revolving in an elliptical orbit.
(Q) A planet revolving in a circular orbit.

Principle of conservation of angular momentum comes in force in which of these?

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 ______.

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?

Answer in brief:

Why is it useful to define the radius of gyration?

Answer in brief:

A uniform disc and a hollow right circular cone have the same formula for their M.I. when rotating about their central axes. Why is it so?

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?

Using the energy conservation, derive the expressions for the minimum speeds at different locations along a vertical circular motion controlled by gravity. Is zero speed possible at the uppermost point? Under what condition/s?

Using energy conservation, along a vertical circular motion controlled by gravity, prove that the difference between the extreme tensions (or normal forces) depends only upon the weight of the objects.

Discuss the necessity of radius of gyration. 

Define radius of gyration

On what factors does radius of gyration depend?

On what factors radius of gyration does not depend?

Can you locate some similarity between the centre of mass and radius of gyration?

What can you infer if a uniform ring and a uniform disc have the same radius of gyration?

Answer in brief:

State the conditions under which the theorems of parallel axes and perpendicular axes are applicable. State the respective mathematical expressions.

Obtain an expression for the torque acting on a rotating body with constant angular acceleration. Hence state the dimensions and SI unit of torque.

State and prove: Law of conservation of angular momentum.

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:

1. The number of revolutions completed by the ant in these 10 seconds.
2. 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:

Part of a racing track is to be designed for curvature of 72m. We are not recommending the vehicles to drive faster than 216 kmph. With what angle should the road be tilted? By what height will its outer edge be, with respect to the inner edge if the track is 10 m wide?

A road is constructed as per the given requirements. The coefficient of static friction between the tyres of a vehicle on this road is 0.8, will there be any lower speed limit? By how much can the upper speed limit exceed in this case?

(Given: r = 72 m, vo = 216 km/h, w = 10 m, θ = 78°4', h = 9.805 m)

During a stunt, a cyclist (considered to be a particle) is undertaking horizontal circles inside a cylindrical well of radius 6.05 m. If the necessary friction coefficient is 0.5, how much minimum speed should the stunt artist maintain? The mass of the artist is 50 kg. If she/he increases the speed by 20%, how much will the force of friction be?

A pendulum consisting of a massless string of length 20 cm and a tiny bob of mass 100 g is set up as a conical pendulum. Its bob now performs 75 rpm. Calculate kinetic energy and increase in the gravitational potential energy of the bob. (Use π² = 10)

A motorcyclist (as a particle) is undergoing vertical circles inside a sphere of death. The speed of the motorcycle varies between 6 m/s and 10 m/s. Calculate the diameter of the sphere of death. How much minimum values are possible for these two speeds?

A metallic ring of mass 1 kg has a moment of inertia 1 kg m² when rotating about one of its diameters. It is molten and remolded into a thin uniform disc of the same radius. How much will its moment of inertia be, when rotated about its own axis.

A big dumb-bell is prepared by using a uniform rod of mass 60 g and length 20 cm. Two identical solid spheres of mass 25 g and radius 10 cm each are at the two ends of the rod. Calculate the moment of inertia of the dumb-bell when rotated about an axis passing through its centre and perpendicular to the length.

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?