Advertisements
Advertisements
Question
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.
Solution
Given:
Msph = 25 g,
Rsph = 10 cm,
Mrod = 60 g,
Lrod = 20 cm
To find:
moment of inertia of the dumb-bell
Solution:
The MI of a solid sphere about its diameter is
`"I"_("sph","CM") = 2/5 "M"_"sph""R"_"sph"^2`
The distance of the rotation axis (transverse symmetry axis of the dumbbell) from the centre of sphere, h = 20cm.
The MI of a solid sphere about the rotation axis,
`"I"_"sph" = "I"_("sph","CM") + "M"_"sph""h"^2`
For the rod, the rotation axis is its transverse symmetry axis through CM.
The MI of a rod about this axis,
Irod = `1/12"M"_"rod""L"_"rod"^2`
Since there are two solid spheres, the MI of the dumbbell about the rotation axis is
i = 2Isph + Irod
`= 2"M"_"sph"(2/5"R"_"sph"^2+"h"^2)+1/12"M"_"rod""L"_"rod"^2`
= `2(25)[2/5(10)^2+(20)^2]+1/12(60)(20)^2`
`= 50 [2/5 (100) + 400] + 1/12(60)(400)`
`= 50 [2/cancel(5) xx cancel(100)^20 + 400] + 1/cancel(12) xx cancel(60)^5 xx (400)`
= 50 [2 × 20 + 400] + 5 × 400
= 50 × 440 + 2000
= 22000 + 2000
= 24000 g cm-2
APPEARS IN
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 ______.
Answer in brief:
Why are curved roads banked?
Do we need a banked road for a two-wheeler? Explain.
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)
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 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.
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 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.
What is a conical pendulum? Obtain an expression for its time period
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 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 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?
