Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
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
संबंधित प्रश्न
Two inclined frictionless tracks, one gradual and the other steep meet at A from where two stones are allowed to slide down from rest, one on each track . Will the stones reach the bottom at the same time? Will they reach there with the same speed? Explain. Given θ1 = 30°, θ2 = 60°, and h = 10 m, what are the speeds and times taken by the two stones?
Figure shows a particle sliding on a frictionless track which terminates in a straight horizontal section. If the particle starts slipping from point A, how far away from the track will the particle hit the ground?
A block of mass m is attached to two unstretched springs of spring constants k1 and k2 as shown in the following figure. The block is displaced towards the right through a distance x and is released. Find the speed of the block as it passes through the mean position shown.
In the following figure shows two blocks A and B, each of mass of 320 g connected by a light string passing over a smooth light pulley. The horizontal surface on which the block Acan slide is smooth. Block A is attached to a spring of spring constant 40 N/m whose other end is fixed to a support 40 cm above the horizontal surface. Initially, the spring is vertical and unstretched when the system is released to move. Find the velocity of the block A at the instant it breaks off the surface below it. Take g = 10 m/s2.
One end of a spring of natural length h and spring constant k is fixed at the ground and the other is fitted with a smooth ring of mass m which is allowed to slide on a horizontal rod fixed at a height h (following figure). Initially, the spring makes an angle of 37° with the vertical when the system is released from rest. Find the speed of the ring when the spring becomes vertical.
Figure following shows a light rod of length l rigidly attached to a small heavy block at one end and a hook at the other end. The system is released from rest with the rod in a horizontal position. There is a fixed smooth ring at a depth h below the initial position of the hook and the hook gets into the ring as it reaches there. What should be the minimum value of h so that the block moves in a complete circle about the ring?
A spring of negligible mass and force constant 5 Nm–1 is compressed by a distance x = 5 cm. A block of mass 200 g is free to leave the end of the spring. If the system is released, what will be the speed of the block when it leaves the spring?
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 ______.
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 ______
A body is falling freely under the action of gravity alone in vacuum. Which of the following quantities remain constant during the fall?
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 ______.
Why is electrical power required at all when the elevator is descending? Why should there be a limit on the number of passengers in this case?
A body falls towards earth in air. Will its total mechanical energy be conserved during the fall? Justify.
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.
