Advertisements
Advertisements
Question
A baloon filled with helium rises against gravity increasing its potential energy. The speed of the baloon also increases as it rises. How do you reconcile this with the law of conservation of mechanical energy? You can neglect viscous drag of air and assume that density of air is constant.
Solution
Because the pulling viscous force of the air on the balloon is ignored, there is
Net Buoyant Force = `Vpg`
The volume of air displaced × net density upward × g
= `V(p_(ar) - p_(He)) g(upward)`
Let a be the upward acceleration on the balloon then
`ma = V(p_(ag) - p_(He)) g`
Where m = mass of the balloon
V = Volume of air displacement by balloon
= Volume of balloon
`p_(air)` = density of air
`p_(He)` = density of helium
`m (dv)/(dt) = V * (p_(ar) - p_(He)) g`
`mdv = V * (p_(air) - p_(He)) g * dt`
Integrating both sides `mv = V * (p_(ar) - p_(He)) g t`
`v = V/m (p_(air) - p_(He)) g t`
KE of balloon = `1/2 mv^2`
`1/2 mv^2 = 1/2 m v^2/m^2 (p_(air) - p_(He))^2g^2t^2`
= `V^2/(2m) (p_(air) - p_(He))^2g^2t^2` ......(ii)
If the balloon rises to a height h, from (i)
`a = V/m (p_(ai) - p_(he)) g`
`h = ut + 1/2 at^2 = 0.t + 1/2 [V/m (p_(ai) - p_(Bb))g]t^2`
∴ `h = V/(2m) (p_(ar) - p_(He)) g t^2`
Rearranging the words of (ii) according to h in (ii) and (iii) (iii)
`1/2 mv^2 = {V/(2m) (P_(air) - p_(He)) g t^2} * V(P_(air) - p_(He)) g`
`1/2 mv^2 = {h} * V (p_(air) - p_(He)) g`
`1/2 mv^2 = V * (p_(at) - p_(He)) gh`
`1/2 mv^2 = Vp_(air) gh - Vp_(He) = gh`
`1/2 mv^2 + p_(He)V gh = p_(av) V gh`
`KE_(balloon) + PE_(ballon)` = Change in PE of air.
As a result, as the balloon rises, an equivalent volume of air falls, increasing the balloon's PE and KE at the expense of the air's PE (which come down).
APPEARS IN
RELATED QUESTIONS
A body is initially at rest. It undergoes one-dimensional motion with constant acceleration. The power delivered to it at time t is proportional to ______.
A person trying to lose weight (dieter) lifts a 10 kg mass, one thousand times, to a height of 0.5 m each time. Assume that the potential energy lost each time she lowers the mass is dissipated.
- How much work does she do against the gravitational force?
- Fat supplies 3.8 x 107J of energy per kilogram which is converted to mechanical energy with a 20% efficiency rate. How much fat will the dieter use up?
You lift a suitcase from the floor and keep it on a table. The work done by you on the suitcase does not depend on
(a) the path taken by the suitcase
(b) the time taken by you in doing so
(c) the weight of the suitcase
(d) your weight
A small heavy block is attached to the lower end of a light rod of length l which can be rotated about its clamped upper end. What minimum horizontal velocity should the block be given so that it moves in a complete vertical circle?
A particle is released from height S from the surface of the Earth. At a certain height, its kinetic energy is three times its potential energy. The height from the surface of the earth and the speed of the particle at that instant are respectively ______
Which of the diagrams shown in figure represents variation of total mechanical energy of a pendulum oscillating in air as function of time?
A mass of 5 kg is moving along a circular path of radius 1 m. If the mass moves with 300 revolutions per minute, its kinetic energy would be ______.
In a shotput event an athlete throws the shotput of mass 10 kg with an initial speed of 1 ms–1 at 45° from a height 1.5 m above ground. Assuming air resistance to be negligible and acceleration due to gravity to be 10 ms–2, the kinetic energy of the shotput when it just reaches the ground will be ______.
A bob of mass m suspended by a light string of length L is whirled into a vertical circle as shown in figure. What will be the trajectory of the particle if the string is cut at
- Point B?
- Point C?
- Point X?
A single conservative force acts on a body of mass 1 kg that moves along the x-axis. The potential energy U(x) is given by U (x) = 20 + (x - 2)2, where x is in meters. At x = 5.0 m the particle has a kinetic energy of 20 J, then the maximum kinetic energy of body is ______ J.
