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प्रश्न
A wooden block of mass m is kept on a piston that can perform vertical vibrations of adjustable frequency and amplitude. During vibrations, we don’t want the block to leave the contact with the piston. How much maximum frequency is possible if the amplitude of vibrations is restricted to 25 cm? In this case, how much is the energy per unit mass of the block? (g ≈ π2 ≈ 10 m/s-2)
उत्तर
Given: A= 0.25 m, g = π2 = 10 m/s2
The acceleration is maximum during vertical oscillations at the top and bottom turning points. The block will just lose contact with the piston when its apparent weight is zero at the top, i.e., when its acceleration is amax = g, downwards.
`|"a"_"max"|` = ω2A = 4π2f2maxA
∴ 4π2f2maxA = g
∴ fmax = `sqrt("g"/π^2. 1/(4"A")`
= `sqrt(10/10. 1/(4(0.25)))` = 1 Hz
This gives the required frequency of the piston.
E = `1/2`mω2A2 = `1/2`m(4π2f2)A2
∴ `"E"/"m"` = 2π2f2A2 = `2(10)(1)^2(1/4)^2`
= `20/16=5/4` = 1.25 J/kg
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