A rounded stone of mass 3.0 kg and radius 0.08 m starts from rest at the top of

Question

A rounded stone of mass 3.0 kg and radius 0.08 m starts from rest at the top of a ramp, inclined 28 degree, and rolls to the bottom without slipping. (For the stone I = 0.7MR^2) The upper end of the ramp is 4.4 m higher than the lower end. Find the linear velocity of the stone when it reaches the bottom of the ramp? (g = 9.8 m/s^2) A) 4.45 m/s B) 7.12 m/s C) 15.4 m/s D) 64.68 m/s E) 92.4 m/s A hot air balloon has a volume of 3359 m^3. The density of the air outside the balloon is 1272 kg/m^3, and the density of the hot air inside is 0.98 kg/m^3. How much weight can the balloon lift? A) 12817.944 N B) 3291.82 N C) 9612 N D) 4360 N E) 9875.46 N

Explanation / Answer

Using law of conservation of energy

energy at the top = energy at the bottom

m*g*h = (0.5*m*v^2) + (0.5*I*w^2)

I = 0.7*M*r^2

w is the angular speed = v/r

then

m*g*h = (0.5*m*v^2)+(0.5*0.7*m*r^2*v^2/r^2)

m cancels

g*h = (0.5*v^2)+(0.5*0.7*v^2)

g*h = 0.5*v^2 + 0.35*v^2

g*h = 0.85*v^2

v = sqrt(g*h/0.85) = sqrt(9.8*4.4/0.85)

v = 7.12 m/sec

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