481 kg metal cylinder is placed inside the top of a plastic tube, the lower end

Question

481 kg metal cylinder is placed inside the top of a plastic tube, the lower end of which is sealed off by an adjustable plunger, and comes to rest some distance above the plunger. The plastic tube has an inner radius of 5.54 mm, and is frictionless. Neither the plunger nor the metal cylinder allow any air to flow around them. If the plunger is suddenly pushed upwards, increasing the pressure between the plunger and the metal cylinder by a factor of 3.03, what is the initial acceleration of the metal cylinder? Assume the pressure outside of the tube is 1.00 atm.

Explanation / Answer

let P1 = 1 atm = 1.013*10^5 pa

cross sectional area of the cyllinder, A = pi*r^2

= pi*(5.54*10^-3)^2

= 9.64*10^-5 m^2

let P2 is the pressure inside,

net force acting on the cyllinder, Fnety = 0

P2*A - P1*A - m*g = 0

P2 = (P1*A + m*g)/A

= P1 + m*g/A

= 1.013*10^5 + 0.481*9.1/(9.64*10^-5)

= 1.467*10^5 pa

when plunger is pushed up,

P2' = 3*P2

= 3*1.467*10^5

= 4.401*10^5 pa

so,

Apply, Fnety = P2*A - P1*A - m*g

m*a = P2*A - P1*A - m*g

a = (P2-P1)*A/m - g

= (4.401 - 1.013)*10^5*9.64*10^-5/0.481 - 9.8

= 58.1 m/s^2

= 0.581*10^2 m/s^2

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