Question

What is the ratio of the linear speeds for the tips of hour and minute hands of a clock if the minute hand is $2.5$ times longer than the hour hand?

• A. $1:30$
• B. $30:1$
• C. $24:5$
• D. $5:24$

Answer 1

The correct option is D $5:24$

Time period for hour hand, $T_h=12hours=12\times 60minutes=12\times 60\times 60seconds=43200seconds{T}_h=\frac{2\pi }{{\omega }_h}=43200s$

where ${\omega }_h$ is angular speed of hour hand

$\therefore {\omega }_h=\frac{2\pi }{43200}rad/s$

Time period for minute hand $T_m=1hour=60minutes=60\times 60s=3600s{T}_m=\frac{2\pi }{{\omega }_m}=3600\Rightarrow {\omega }_m=\frac{2\pi }{3600}rad/s\frac{linearspeedofhourhand}{linearspeedofminutehand}=\frac{r_h{\omega }_h}{r_m{\omega }_m}$

$=\frac{2.5{\omega }_h}{{\omega }_m}=2.5\times \frac{2\pi }{43200}\times \frac{3600}{2\pi }$

$=2.5\times \frac{36}{432}=\frac{5}{2}\times \frac{1}{12}=\frac{5}{24}$