Advertisements
Advertisements
प्रश्न
Give the location of the centre of mass of a
- sphere,
- cylinder,
- ring, and
- cube,
each of uniform mass density. Does the centre of mass of a body necessarily lie inside the body?
उत्तर
For objects with uniform mass density, such as a sphere, cylinder, ring, and cube, the centre of mass is located at their respective geometric centres. This is due to the symmetrical distribution of mass throughout these objects. However, the centre of mass of a body does not necessarily have to lie within the physical material of the body. A notable example of this is a circular ring, where the centre of mass is at the geometric centre of the ring but in an area where there is no physical mass.
APPEARS IN
संबंधित प्रश्न
A circular plate of diameter d is kept in contact with a square plate of edge d as show in figure. The density of the material and the thickness are same everywhere. The centre of mass of the composite system will be 
Consider a gravity-free hall in which a tray of mass M, carrying a cubical block of ice of mass m and edge L, is at rest in the middle. If the ice melts, by what distance does the centre of mass of "the tray plus the ice" system descend?
During a heavy rain, hailstones of average size 1.0 cm in diameter fall with an average speed of 20 m/s. Suppose 2000 hailstones strike every square meter of a 10 m × 10 m roof perpendicularly in one second and assume that the hailstones do not rebound. Calculate the average force exerted by the falling hailstones on the roof. Density of a hailstone is 900 kg/m3.
A railroad car of mass M is at rest on frictionless rails when a man of mass m starts moving on the car towards the engine. If the car recoils with a speed v backward on the rails, with what velocity is the man approaching the engine?
A block of mass 2.0 kg moving 2.0 m/s collides head on with another block of equal mass kept at rest. (a) Find the maximum possible loss in kinetic energy due to the collision. (b) If he actual loss in kinetic energy is half of this maximum, find the coefficient of restitution.
The axis of rotation of a purely rotating body
(a) must pass through the centre of mass
(b) may pass through the centre of mass
(c) must pass through a particle of the body
(d) may pass through a particle of the body.
A round object of mass M and radius R rolls down without slipping along an inclined plane. The frictional force, ______
Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass:
- Show pi = p’i + miV Where pi is the momentum of the ith particle (of mass mi) and p′ i = mi v′ i. Note v′ i is the velocity of the ith particle relative to the centre of mass. Also, prove using the definition of the centre of mass `sum"p""'"_"t" = 0`
-
Show K = K′ + 1/2MV2
where K is the total kinetic energy of the system of particles, K′ is the total kinetic energy of the
system when the particle velocities are taken with respect to the centre of mass and MV2/2 is the
kinetic energy of the translation of the system as a whole (i.e. of the centre of mass motion of the
system). The result has been used in Sec. 7.14. - Show where `"L""'" = sum"r""'"_"t" xx "p""'"_"t"` is the angular momentum of the system about the centre of mass with
velocities taken relative to the centre of mass. Remember `"r"_"t" = "r"_"t" - "R"`; rest of the notation is the standard notation used in the chapter. Note L′ and MR × V can be said to be angular momenta, respectively, about and of the centre of mass of the system of particles. - Show `"dL"^"'"/"dt" = ∑"r"_"i"^"'" xx "dP"^"'"/"dt"`
Further show that `"dL"^'/"dt" = τ_"ext"^"'"`
Where `"τ"_"ext"^"'"` is the sum of all external torques acting on the system about the centre of mass.
(Hint: Use the definition of centre of mass and third law of motion. Assume the internal forces between any two particles act along the line joining the particles.)
For which of the following does the centre of mass lie outside the body?
Find the centre of mass of a uniform (a) half-disc, (b) quarter-disc.
