Vertical circular motion — minimum speed conditions

1.2.18 · Physics › Newton's Laws & Dynamics

WHY does a minimum speed even exist?

Loop ke top pe, circle ka center ball ke neeche hota hai. Toh "center ki taraf" = neeche. Gravity ( $m g$ , hamesha neeche) aur tension/normal force ( $T$ ya $N$ , center ki taraf pull karte hue = neeche) dono wahan neeche point karte hain.

$T + m g = \frac{m v _{top}^{2}}{r}$

HOW to derive the minimum speed at the top

Yeh step kyun? $T = 0$ set karna physical edge ko capture karta hai: string slack hone ki verge pe hai. Isse neeche, koi real configuration ise circular nahi rakh sakti.

HOW to derive the minimum speed at the BOTTOM

Poore loop ko survive karne ke liye tumhe bottom mein itni fast enter karna hoga ki top pe abhi bhi $v_{top,min}$ mile. Energy conservation use karo (string tension koi kaam nahi karti — woh motion ke perpendicular hai).

Figure — Vertical circular motion — minimum speed conditions

Tension at the bottom (the "5mg" result)

Worked Examples

Common Mistakes

Active Recall

Recall Top pe minimum speed kya hai, aur ise define karne wali physical condition kya hai?

$v_{top,min} = g r$ , tension (ya normal force) $= 0$ se define hota hai, yaani gravity akele centripetal force supply karti hai.

Recall Bottom speed

$5 g r$ kyun hai aur $g r$ kyun nahi? Energy conservation: height $2 r$ climb karna $\frac{1}{2} v^{2}$ terms mein $2 g r$ costa hai (yaani $v^{2}$ mein $4 g r$ ). $v_{bot}^{2} = v_{top}^{2} + 4 g r = g r + 4 g r = 5 g r$ .

Recall String ki jagah rigid rod ke liye kya badalta hai?

Rod push kar sakta hai, toh $T$ negative ho sakta hai; minimum-speed condition $v_{top} \geq 0$ ban jaati hai, jo $v_{bot,min} = 2 g r$ deti hai.

Recall (Feynman) Ek 12-saal ke bacche ko explain karo kyun top pe slow hone se ball girta hai.

Socho tum ek ball ko string pe tez apne sar ke upar swing kar rahe ho. String tight rehti hai kyunki ball seedha fly karna chahta hai lekin string use circle mein kheenchti hai. Agar tum bilkul top pe bahut slowly swing karo, toh gravity pehle se ball ko utna neeche kheeench rahi hai jitna curve karne ke liye chahiye se zyada — toh string floppy ho jaati hai aur ball circle karne ki jagah sirf gir jaata hai. "Bas itna fast" speed woh hai jab gravity exactly utna hi pull karti hai jitna circle ko chahiye.

Flashcards

Minimum speed at the top of a vertical circle (string)

v_{top} = g r

(set tension

= 0

)

Condition that defines minimum speed at top

Tension/normal force

= 0

; gravity alone gives centripetal force

Minimum speed at bottom to complete the loop

v_{bot} = 5 g r

Speed at the horizontal side (height

r

) for minimum loop

v_{side} = 3 g r

Ratio

v_{b o t}^{2} : v_{s i d e}^{2} : v_{t o p}^{2}

5 : 3 : 1

(in units of

g r

)

Tension at the bottom for the minimum-speed loop

6 m g

Tension at the top for the minimum-speed loop

0

Difference

T_{b o t} - T_{t o p}

for any speed in a vertical circle

6 m g

Why bottom needs more speed than

g r

Energy to climb height

2 r

adds

4 g r

v^{2}

Minimum top speed for a rigid ROD

0

(rod can push, no slack condition)

Direction of net centripetal force at the top

Downward (toward center, which is below)

What does

\frac{m v ^{2}}{r}

represent

The required net inward force, not a separate force

Connections

Centripetal force and acceleration
Newton's Second Law — net force form
Conservation of mechanical energy
Tension and constraint forces
Normal force in circular tracks
Banked curves and horizontal circular motion
Free-body diagrams