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प्रश्न
A pendulum consisting of a massless string of length 20 cm and a tiny bob of mass 100 g is set up as a conical pendulum. Its bob now performs 75 rpm. Calculate kinetic energy and increase in the gravitational potential energy of the bob. (Use π2 = 10)
उत्तर
Data: L = 0.2 m, m = 0.1 kg, n = `75/60 = 5/4`rps,
g = 10 m/s2, π2 = 10,
T = `1/"n" = 4/5 "s" = 0.8 "s"`
T = `2pi sqrt(("L cos"theta)/"g")`
∴ T2 = `4pi^2 ("L cos"theta)/"g"`
∴ h = L cos θ = `"gT"^2/(4pi^2)`
`= ((10)(0.8)^2)/(4(10))`
= 0.16 m ...(1)
∴ cos θ = `0.16/0.2 = 0.8`
∴ θ = cos-1 0.8 = 36.87° = 36°5'
v2 = rg tan θ = (L sin θ)(g) tan 36.87°
= (0.12)(10)(0.7500)
= 0.9
The KE of the bob `= 1/2 "mv"^2`
`= 1/2(0.1)(0.9)`
= 0.045 J
The increase in gravitational PE,
Δ PE = mg (L - h)
= (0.1)(10)(0.2 - 0.16)
= 0.04 J
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