Question

A particle describes a horizontal circle on smooth inner surface of a cone as shown in figure. If the height of the circle above the vertex is 10 cm. The speed of the particle is (g, acceleration due to gravity = $10 m/s^2$)

• A. 2 m/s
• B. 1.5 m/s
• C. 1 m/s
• D. 0.5 m/s

Hint:

For a cone, the orbital speed $v$ depends only on the height $h$ from the vertex, not the mass.

Answer 1

The Correct Option is C

Step 1: Force Balance
$N \cos \alpha = mg$ and $N \sin \alpha = \frac{mv^2}{r}$.
$\tan \alpha = \frac{v^2}{rg}$, where $\alpha$ is the semi-vertical angle.
Step 2: Geometric Relation
From geometry, $\tan \alpha = \frac{r}{h}$.
Step 3: Combining Equations
$\frac{r}{h} = \frac{v^2}{rg} \Rightarrow v^2 = gh$
Step 4: Calculation
$h = 10 \text{ cm} = 0.1 \text{ m}$
$v = \sqrt{10 \times 0.1} = \sqrt{1} = 1 \text{ m/s}$
Final Answer:(C)