Circular motion — horizontal circle, vertical circle, conical pendulum types

Question

What are the different types of circular motion problems — horizontal circle, vertical circle, and conical pendulum — and how do we set up the equations for each?

Solution — Step by Step

A mass moving in a horizontal circle at constant speed has centripetal acceleration directed toward the centre:

a_c = \frac{v^2}{r} = \omega^2 r

The net force toward the centre provides this acceleration:

F_{\text{net, radial}} = \frac{mv^2}{r}

For a car on a banked road (angle $\theta$ , no friction):

\tan\theta = \frac{v^2}{rg}

In a vertical circle (like a ball on a string), gravity adds to or subtracts from the centripetal force depending on position.

At the top of the circle:

mg + T_{top} = \frac{mv_{top}^2}{r}

At the bottom of the circle:

T_{bottom} - mg = \frac{mv_{bottom}^2}{r}

For the string to remain taut at the top, $T_{top} \geq 0$ , which gives the minimum speed:

v_{top, \text{min}} = \sqrt{gr}

Using energy conservation from bottom to top:

v_{bottom, \text{min}} = \sqrt{5gr}

A bob on a string traces a horizontal circle while the string sweeps out a cone. The setup:

String length: $L$ , makes angle $\theta$ with vertical
Radius of circle: $r = L\sin\theta$

Vertical equilibrium: $T\cos\theta = mg$

Radial (centripetal): $T\sin\theta = \frac{mv^2}{r}$

Dividing: $\tan\theta = \frac{v^2}{rg}$

Time period:

T_{\text{period}} = 2\pi\sqrt{\frac{L\cos\theta}{g}}

The time period of a conical pendulum depends on $L\cos\theta$ (the vertical height from the circle to the support), not on the mass. This is similar to a simple pendulum but with $L$ replaced by $L\cos\theta$ .

Feature	Horizontal circle	Vertical circle	Conical pendulum
Speed	Constant	Varies with position	Constant
Gravity role	No role in centripetal force	Adds/subtracts from centripetal force	Balanced by vertical tension component
Critical condition	None usually	$v_{min}$ at top	Angle determines speed

flowchart TD
    A["Circular Motion Problem"] --> B{"Plane of circle?"}
    B -->|"Horizontal"| C{"String at angle?"}
    B -->|"Vertical"| D["Vertical circle"]
    C -->|"Yes, cone shape"| E["Conical pendulum"]
    C -->|"No"| F["Horizontal circle"]
    D --> G["Use energy conservation + F=mv2/r at each point"]
    E --> H["Resolve tension: T cos theta = mg, T sin theta = mv2/r"]
    F --> I["Net radial force = mv2/r"]

Why This Works

In all circular motion, Newton’s second law applied along the radial direction gives us the centripetal force equation. The only difference between types is which forces contribute to (or oppose) the centripetal acceleration and whether the speed changes during the motion.

In horizontal circles, gravity acts perpendicular to the plane of motion, so it does not affect centripetal force. In vertical circles, gravity is in the plane of motion, so it directly adds or subtracts from the radial force balance.

Alternative Method

For vertical circle problems, the energy method is often faster than force analysis at every point. Write energy conservation between bottom and any angle $\theta$ , then substitute into the force equation at that angle. This gives tension as a function of $\theta$ in one step.

Common Mistake

In vertical circle problems, students use $v = \sqrt{gr}$ as the minimum speed at the bottom. That is the minimum speed at the top. At the bottom, the minimum speed for completing the circle is $v = \sqrt{5gr}$ . The factor of 5 comes from energy conservation between top and bottom (a height difference of $2r$ ). This error appeared as a distractor in JEE Main 2024 January Shift 2.

Question

Solution — Step by Step

Why This Works

Alternative Method

Common Mistake

Try These Next