Question

a cylinder is fitted with a piston, beneath which is a spring, as in the drawing. the cylinder is open to the air at the top. friction is absent. the spring constant of the spring is 4200 n/m. the piston has a negligible mass and a radius of 0.023 m. (a) when the air beneath the piston is completely pumped out, how much does the atmospheric pressure cause the spring to compress? (b) how much work does the atmospheric pressure do in compressing the spring?

Answer 1

0.39 cm is the compression in the spring by atmospheric pressure. Work done by the atmospheric pressure is 0.031J in compressing the spring.

a) When the air beneath the piston is completely pumped out, then formula for the atmospheric pressure causing the spring to compress will be: p°A = kx

where, x= compression, A= area, p°= atmospheric pressure and k= spring constant

Now, p°A = kx

x = p°A/x

x = p° (πr²)/k

Putting the values: x= {(1.0 × 10⁵) × π × (0.023)²}/4200

x= 0.0039 meter

Thus, x= 0.39 cm is the compression in the spring.

b) work done by the atmospheric pressure to compress the spring will be: W=(1/2)kx²

where, w= work done, k= spring constant and x= compression

Now, W=(1/2)kx²

W= (1/2) × 4200 × (0.0039)²

W= 0.031 Joule

Thus, work done by the atmospheric pressure is 0.031J.

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