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## Problems on Rigid bodies

A rigid body is defined as a body in which distance between the two particles remains constant.

The following are the analogues parameters of rotational and translational motion:-

 Translational motion Rotational motion Quantity Definition Meaning Quantity Definition Meaning Position x Coordinate of x-axis in reference frame Angular position Angle made with reference line Linear Velocity Distance covered in unit time Angular velocity Angle swept in unit time Linear acceleration Change in the velocity in unit time Angular acceleration Change in the angular velocity in unit time Mass M Rotational Inertia Newton’s second law (Force) F = ma Rate of change of linear momentum Newton’s second law (Torque) Rate of change of angular momentum Work Work Kinetic Energy Kinetic Energy Power Power Diagrammatic representation of above quantities: The kinematical equations for rotating body:-

The equations hold good when acceleration is uniform. Each formula has one variable which is not required. Observe the formulae and note down the missing variables too.

Problems:-

Q1)            The following is the equation of a rotating body. Using the data given calculate

a)      ω (t) between t=10sec and t=15 sec.

b)      α (t) between t=20 sec and 27 sec.

Q2)            Our Sun is from the centre of our Milky Way galaxy and is moving in a circle around that centre at a speed 250km/s.

a)      How long does it take to make one revolution about galactic centre?

b)      How many revolutions has sun completed since it was formed about ago?

Q3)            At 7:34 A.M. on June 30, 1908 a huge explosion has occurred above central Siberia at latitude 61 degrees North and longitude 102 degrees east; the fireball thus created was the brightest flash seen by anyone before the advent of nuclear weapons. The “Tunguska Event”, which according to one chance witness “covered an enormous part of sky” was probably the explosion of a stony asteroid about 140 m wide.

a)      Considering only the earth’s rotation determine how much later the asteroid would have had to arrive to put the explosion above Helsinki at longitude 25 degrees East. This would have obliterated the city.

b)      If the asteroid been a metallic asteroid, it could have reached the earth’s surface. How much later would sun an asteroid have had to arrive to put the impact in the Atlantic ocean at longitude 20 degrees west? (Resulting Tsunamis would have wiped out costal civilizations on both sides of Atlantic).

Q4)            An astronautic is being tested in a centrifuge the centrifuge ahs a radius of 10m and in starting rotates according to where t is in seconds and theta is in radians. When t= 5 seconds what are astronauts

a)      Angular velocity

b)      Linear speed

c)      Tangential acceleration

Q5)            The oxygen molecule has a total mass of kg. and a rotational inertia of km.m2 about an axis through the centre of line joining the atoms and perpendicular to that line. Suppose that such a molecule of gas hads a speed of 500m/s and that it’s rotational kinetic energy is two thirds of its translational kinetic energy. Find its angular velocity.