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प्रश्न
Define centre of mass.
उत्तर
The center of mass of a body is defined as a point where the entire mass of the body appears to be concentrated.
APPEARS IN
संबंधित प्रश्न
In an elastic collision
A square plate of edge d and a circular disc of diameter d are placed touching each other at the midpoint of an edge of the plate as shown in figure. Locate the centre of mass of the combination, assuming same mass per unit area for the two plates.
Calculate the velocity of the centre of mass of the system of particles shown in figure.
A car of mass M is at rest on a frictionless horizontal surface and a pendulum bob of mass m hangs from the roof of the cart. The string breaks, the bob falls on the floor, makes serval collisions on the floor and finally lands up in a small slot made in the floor. The horizontal distance between the string and the slot is L. Find the displacement of the cart during this process.
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.
A round object of mass M and radius R rolls down without slipping along an inclined plane. The frictional force, ______
Which of the following has maximum momentum?
The ratio of weights of a man inside a lift when it is stationary and when it is going down with a uniform acceleration 'a' is 3 : 2. The value of 'a' will be ______.
(a< g, g = acceleration due to gravity)
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.)
Figure shows a lamina in x-y plane. Two axes z and z ′ pass perpendicular to its plane. A force F acts in the plane of lamina at point P as shown. Which of the following are true? (The point P is closer to z′-axis than the z-axis.)
- Torque τ caused by F about z axis is along `-hatk`.
- Torque τ′ caused by F about z′ axis is along `-hatk`.
- Torque τ caused by F about z axis is greater in magnitude than that about z axis.
- Total torque is given be τ = τ + τ′.
