Question

The lengths of needles of hour and minute of a clock are 4 cm and 6 cm respectively.

(i) Find the ratio of linear velocities of tips of the needles.

(ii) If the length of hour needle be 3 cm, then what will be the above ratio?

Answer 1

(i) r₁ = 4 cm, T₁ = 12 hr

r₂ = 6 cm, T₂ = 1 hr

v₁ : v₂ = ?

\(\because\ v = r\omega = r.\frac{2\pi}{T}\)

\(\therefore\ \frac{v_1}{v_2} = \frac{r_1\frac{2\pi}{T_1}}{r_2\frac{2\pi}{T_2}} = \frac{r_1}{r_1}\times\frac{T_2}{T_1} = \frac{4\ cm}{6\ cm}\times\frac{1hr}{12hr}\)

\( = \frac{1}{18}\therefore\ V_1:V_2 = 1:18\)

(ii) If r₁ = 3 cm, then

\(\frac{v_1}{v_2} = \frac{r_1}{r_2}\times\frac{T_2}{T_1} = \frac{3\ cm}{6\ cm}\times\frac{1hr}{12hr} = \frac{1}{24}\)

∴ v₁ : v₂ = 1 : 24