SATHEE: Laws of Motion 4 Question 10

Laws of Motion 4 Question 10

10. A hemispherical bowl of radius $R = 0.1 m$ is rotating about its own axis (which is vertical) with an angular velocity $ω$ . A particle of mass $10^{- 2} k g$ on the frictionless inner surface of the bowl is also rotating with the same $ω$ . The particle is at a height $h$ from the bottom of the bowl.

$(1993, 3 + 2 M)$

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Answer:

Correct Answer: 10. (a) $h = R - \frac{g}{ω^{2}}, ω_{min} = \sqrt{\frac{g}{R}} = 9.89 r a d / s$

(b) $9.8 \times 10^{- 3} m / s^{2}$

Solution:

Given, $R = 0.1 m, m = 10^{- 2} k g$

(a) FBD of particle in ground frame of reference is shown in figure. Hence,

and $N \sin θ = m r ω^{2}$

Dividing Eq. (ii) by Eq. (i), we obtain

$\begin{aligned} \tan θ & = \frac{r ω^{2}}{g} or \frac{r}{R - h} = \frac{r ω^{2}}{g} \\ or ω^{2} & = \frac{g}{R - h} \end{aligned}$

This is the desired relation between $ω$ and $h$ . From Eq. (iii),

$h = R - \frac{g}{ω^{2}}$

For non-zero value of $h$ ,

$R > \frac{g}{ω^{2}} or ω > \sqrt{g / R}$

Therefore, minimum value of $ω$ should be

$\begin{aligned} ω_{min} & = \sqrt{\frac{g}{R}} = \sqrt{\frac{9.8}{0.1}} r a d / s \\ ω_{min} & = 9.89 r a d / s \end{aligned}$

(b) Eq. (iii) can be written as $h = R - \frac{g}{ω^{2}}$

If $R$ and $ω$ are known precisely, then $Δ h = - \frac{Δ g}{ω^{2}}$

or $Δ g = ω^{2} Δ h$ (neglecting the negative sign)

$(Δ g)_{min} = {(ω_{min})}^{2} Δ h, (Δ g)_{min} = 9.8 \times 10^{- 3} m / s^{2}$

पिछला

Laws of Motion 4 Question 10

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

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परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

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Laws of Motion 4 Question 10

10. A hemispherical bowl of radius R=0.1m is rotating about its own axis (which is vertical) with an angular velocity ω. A particle of mass 10−2kg on the frictionless inner surface of the bowl is also rotating with the same ω. The particle is at a height h from the bottom of the bowl.

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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