A shell of mass m is at rest initially. It explodes into three fragments having masses in the ratio 2 : 2 : 1. The fragments having equal masses fly off along mutually perpendicular direction with speed v. What will be the speed of the third (lighter) fragment ?
A small spherical ball of mass m slides without friction from the top of a hemisphere of radius R. At what height will the ball lose contact with surface of the sphere ?
We know that if the ball looses contact at B, then from conservation of energy, we get
So, we get v2 = 2gR (1 − cos θ) ...... (i)
Hence at point B, we have
So, when the ball looses contact, the normal reaction becomes equal to zero, ie, N = 0
Hence,
This along with Eqn. (i), we get
v2 = Rg cos θ = 2gR (1 − cos θ)
Hence, we have
2 − 2 cos θ = cos θ
So, cos θ =
Hence the height from the ground would be equal to
h = R cos θ =
Two identical cylindrical vessels, with their bases at the same level, each contain a liquidof density ρ. The height of liquid in one vessel in h1 and that in the other is h2. The area of either base is A. What is the work done by gravity in equalizing the levels when the vessels are interconnected ?
A battery of emf E and internal resistance r is connected across a pure resistive device (such as an electric heater) of resistance R. Prove that the power output of the device will be maximum if R = r.
A radioactive isotope X with half life 1.5x 109 yr decays into a stable nucleus Y. A rock sample contains both elements X and Y in ratio 1 : 15. Find the age of the rock.