Question 402: Properties of Matter

Question

A table tennis ball has radius $(3 / 2) 	imes 10^{-2} \mathrm{~m}$ and mass $(22 / 7) 	imes 10^{-3} \mathrm{~kg}$. It is slowly pushed down into a swimming pool to a depth of $d=0.7 \mathrm{~m}$ below the water surface and then released from rest. It emerges from the water surface at speed $v$, without getting wet, and rises up to a height $H$. Which of the following option(s) is(are) correct?
[Given: $\pi=22 / 7, g=10 \mathrm{~m} \mathrm{~s}^{-2}$, density of water $=1 	imes 10^3 \mathrm{~kg} \mathrm{~m}^{-3}$, viscosity of water $=1 	imes 10^{-3} \mathrm{~Pa}$-s.]

Accepted Answer

$$\begin{aligned}
& 	ext { (A) } w_{	ext {all }}=k_f-k_i=0 \
& w_g+w_B+w_v+w_{	ext {ext }}=0 \
& \mathrm{mgd}-ho_w \cdot v \cdot g d-6 \pi \eta r v d+w_{	ext {ext }}=0 \
& \quad(	ext { slowly } v=0)
\end{aligned}$$
$$\begin{aligned}
& \mathrm{w}_{	ext {ext }}=ho_{\mathrm{w}} \operatorname{vgd}-\mathrm{mgd}=\left(1000 	imes \frac{4}{3} 	imes \frac{22}{7} 	imes\left(\frac{3}{2} 	imes 10^{-2}ight)^3-\frac{22}{7} 	imes 10^{-3}ight) \mathrm{gd} \
& \mathrm{w}_{	ext {ext }}=\frac{22}{7} 	imes 10^{-3}\left[\frac{9}{2}-1ight] 	imes 10 	imes 0.7=\frac{22}{7} 	imes 10^{-3} 	imes \frac{7}{2} 	imes 7 \
& \mathrm{~W}_{	ext {ext }}=77 	imes 10^{-3} \mathrm{~J}=0.077 \mathrm{~J}
\end{aligned}$$
$$\begin{aligned}
& 	ext { (B) } \mathrm{w}_{\mathrm{g}}+\mathrm{w}_{\mathrm{B}}=\mathrm{k}_{\mathrm{f}}-\mathrm{k}_{\mathrm{i}} \quad\left(\mathrm{k}_{\mathrm{i}}=0ight) \
& \frac{1}{2} 	imes \frac{22}{7} 	imes 10^{-3} \mathrm{v}^2=77 	imes 10^{-3} \
& \mathrm{v}^2=\frac{77 	imes 7 	imes 2}{22} \
& \mathrm{v}=7 \mathrm{~m} / \mathrm{s}
\end{aligned}$$
(C) $$\mathrm{H}=\frac{\mathrm{v}^2}{2 \mathrm{~g}}=\frac{49}{20}=2.45 \mathrm{~m}$$

Answer

The work done in pushing the ball to the depth $d$ is $0.077 \mathrm{~J}$.

Answer

If we neglect the viscous force in water, then the speed $v=7 \mathrm{~m} / \mathrm{s}$.

Answer

If we neglect the viscous force in water, then the height $H=1.4 \mathrm{~m}$.

Answer

The ratio of the magnitudes of the net force excluding the viscous force to the maximum viscous force in water is $500 / 9$.

Properties of Matter

Choose the correct answer: