Vertical Circle and Pseudo Forces

A stone whirled in a vertical circle does not move uniformly — gravity speeds it up on the way down and slows it on the way up. The string tension also varies dramatically. To analyze accelerating frames cleanly (like inside a turning car) we introduce pseudo forces.

Concept

Vertical circle. A body of mass $m$ tied to a string of length $L$ moves in a vertical circle. At a general angle $\theta$ from the lowest point, write Newton's second law along the radius (toward the centre):

$T - mg\cos\theta = \frac{mv^2}{L}$

(at angle $\theta$ measured from the bottom, with $\cos\theta$ being the component of weight along the string toward the centre when at the bottom and away at the top).

At the top of the circle: $T_{\text{top}} + mg = mv_{\text{top}}^2/L$. The string can only pull (not push), so $T_{\text{top}} \geq 0$. The minimum speed at the top occurs when $T_{\text{top}} = 0$:

$v_{\text{top, min}} = \sqrt{gL}$

Minimum speed at the bottom (so the stone barely completes the loop): by energy conservation,

$\tfrac{1}{2}m v_{\text{bot}}^2 = \tfrac{1}{2}m v_{\text{top}}^2 + mg(2L)$

With $v_{\text{top}}^2 = gL$,

$v_{\text{bot, min}} = \sqrt{5gL}$

Tension difference (any complete vertical circle): $T_{\text{bot}} - T_{\text{top}} = 6mg$.

Pseudo forces. Newton's laws hold only in inertial frames. In a frame accelerating with acceleration $\vec{a}_0$, to apply $\vec{F}=m\vec{a}$ we must add a pseudo force $\vec{F}_{\text{pseudo}} = -m\vec{a}_0$ on each body. Centrifugal force (in a rotating frame) and the Coriolis force are pseudo forces.

Derivation

Minimum speed at the top. At the top, both gravity and tension point downward (toward the centre):

$T + mg = \frac{mv_{\text{top}}^2}{L}$

For the string to remain taut, $T \geq 0$, hence

$\frac{mv_{\text{top}}^2}{L} \geq mg \quad \Longrightarrow \quad v_{\text{top}}^2 \geq gL$

Minimum speed at the bottom. Conservation of mechanical energy between bottom and top of the circle (height difference $2L$):

$\tfrac{1}{2}mv_{\text{bot}}^2 = \tfrac{1}{2}mv_{\text{top}}^2 + 2mgL$

Set $v_{\text{top}}^2 = gL$:

$v_{\text{bot}}^2 = gL + 4gL = 5gL$

$\boxed{v_{\text{bot, min}} = \sqrt{5gL}}$

Tension difference. Bottom: $T_{\text{bot}} - mg = mv_{\text{bot}}^2/L$. Top: $T_{\text{top}} + mg = mv_{\text{top}}^2/L$. Subtract:

$T_{\text{bot}} - T_{\text{top}} = 2mg + \frac{m(v_{\text{bot}}^2 - v_{\text{top}}^2)}{L} = 2mg + \frac{m \cdot 4gL}{L} = 6mg$

Worked Example

A 0.5 kg stone is tied to a 1 m string and rotated in a vertical circle. (a) What is the minimum speed at the top so the string stays taut? (b) What is the tension at the bottom in that limiting case? Take $g = 10 \, \text{m/s}^2$.

Solution:

(a) $v_{\text{top, min}} = \sqrt{gL} = \sqrt{10 \times 1} = \sqrt{10} \approx 3.16 \, \text{m/s}$.

(b) At the bottom, $v_{\text{bot}}^2 = 5gL = 50 \, \text{m}^2/\text{s}^2$. Tension:

$T_{\text{bot}} = mg + \frac{mv_{\text{bot}}^2}{L} = (0.5)(10) + \frac{(0.5)(50)}{1} = 5 + 25 = 30 \, \text{N}$

The tension at the bottom is six times the weight (5 N) plus the top tension (zero) — confirming $T_{\text{bot}} - T_{\text{top}} = 6mg$.

Common Confusions

• Centrifugal force is real in the rotating frame. In an inertial frame it doesn't exist; in a co-rotating frame you must include it to apply $\vec{F}=m\vec{a}$.
• String tension can never be negative. If math says $T < 0$, the string goes slack and the body follows projectile motion.
• Energy conservation is the cleanest tool for relating speeds at different points in a vertical circle (since gravity is conservative).
• Pseudo forces are not Newtonian forces — they have no reaction partner. They are mathematical patches for non-inertial frames.

Key Takeaways

• Minimum speed at top of vertical circle: $v_{\text{top}} = \sqrt{gL}$.
• Minimum speed at bottom: $v_{\text{bot}} = \sqrt{5gL}$.
• Tension difference between bottom and top: $6mg$, independent of speed.
• Pseudo force in a frame with acceleration $\vec{a}_0$ is $-m\vec{a}_0$.
• Centrifugal and Coriolis forces are pseudo forces in rotating frames.