Circular Motion Tool:The path is circular, so the direction of velocity keeps changing even if the speed is constant. This change in direction produces acceleration towards the centre of the circle.यह टूल एक इंटरएक्टिव फिजिक्स सिमुलेटर है जिसे विशेष रूप से समान वृत्तीय गति (Uniform Circular Motion) और औसत त्वरण (Average Acceleration) की अवधारणा को समझाने के लिए डिज़ाइन किया गया है।Circular Motion Tool

There are two main types:

• Uniform Circular Motion (UCM): Speed is constant, but velocity changes continuously (only direction changes). Example: Earth revolving around the Sun (approximately), blades of a fan, car moving at constant speed on a circular track.
• Non-Uniform Circular Motion: Both speed and direction change. Example: A car speeding up or slowing down while turning.

1. Angular Quantities (Easier for Circular Motion)

We use angular quantities because they simplify calculations for circular paths.

• Angular Displacement (θ): The angle subtended by the position vector at the centre. Unit: radian (rad). Relation: θ = s / r (where s = arc length, r = radius). 2π rad = 1 revolution = 360°.
• Angular Velocity (ω): Rate of change of angular displacement. ω = dθ/dt (instantaneous) or ω = Δθ/Δt (average). Unit: rad/s. For uniform motion: ω = 2πf = 2π/T (f = frequency = revolutions per second, T = time period = time for one revolution).
• Angular Acceleration (α): Rate of change of angular velocity. α = dω/dt. Unit: rad/s². In UCM, α = 0.Circular Motion Tool

Kinematic Equations (for constant α, similar to linear motion): ω = ω₀ + αt θ = ω₀t + (1/2)αt² ω² = ω₀² + 2αθ

2. Relation Between Linear and Angular Quantities

• Linear displacement: s = rθ
• Linear (tangential) velocity: v = rω
• Tangential acceleration: a_t = rα (this changes the speed)

Even in UCM (α = 0, a_t = 0), there is acceleration due to change in direction.

3. Centripetal Acceleration (Radial Acceleration)

This acceleration is directed towards the centre of the circle and is responsible for changing the direction of velocity. a_c = v² / r = rω² = vω

• In UCM: Total acceleration = a_c (only radial).
• In non-uniform circular motion: Total acceleration a = √(a_t² + a_c²) Direction: tan⁻¹(a_t / a_c) from the radial direction.Circular Motion Tool

4. Centripetal Force

The net force required to keep the object in circular motion (towards the centre). F_c = m a_c = m v² / r = m r ω²

Important points:

• Centripetal force is not a new type of force. It is provided by existing forces like tension, friction, gravity, etc.
• It is always perpendicular to velocity → Work done by centripetal force = 0.
• Centrifugal force is a pseudo-force (appears only in non-inertial rotating frames) and acts outward.Circular Motion Tool

5. Applications and Special Cases

a) Banking of Roads (no friction case) For safe turning at speed v: tan θ = v² / (r g) (where θ = banking angle)

b) Conical Pendulum A mass tied to a string swinging in a horizontal circle. tan θ = v² / (r g) or T = 2π √(l cos θ / g) (θ = semi-vertical angle, l = length of string)

c) Motion in Vertical Circle

• At lowest point: T – mg = m v_b² / r → T = mg + m v_b² / r (maximum tension)
• At highest point: T + mg = m v_t² / r → T = m v_t² / r – mg (minimum tension)
• Minimum speed at top (to complete circle): √(gr)
• Minimum speed at bottom (to complete circle with string): √(5gr)

d) Other Examples

• Satellites (gravitational force provides F_c)
• Electron in magnetic field
• Washing machine dryer (centrifugal effect)

Important Note: In UCM, kinetic energy is constant but linear momentum changes (direction changes).

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5 Numerical Problems with Solutions (Class 11 level, SI units, g = 10 m/s²)

Problem 1 A particle moves in a circle of radius 0.5 m with a linear speed of 10 m/s. Find (a) angular velocity, (b) centripetal acceleration, (c) time period.

Solution (a) ω = v / r = 10 / 0.5 = 20 rad/s (b) a_c = v² / r = (10)² / 0.5 = 200 m/s² (c) T = 2π / ω = 2π / 20 = π/10 s ≈ 0.314 s

Problem 2 An object is revolving in a horizontal circle of radius 2 m with angular velocity 5 rad/s. Calculate its linear speed and centripetal acceleration.

Solution Linear speed v = rω = 2 × 5 = 10 m/s Centripetal acceleration a_c = rω² = 2 × (5)² = 50 m/s² (Alternatively: a_c = v² / r = 100 / 2 = 50 m/s²)

Problem 3 A car is moving on a banked road of radius 50 m with banking angle 30°. Find the maximum speed without relying on friction. (g = 10 m/s²)

Solution tan θ = v² / (r g) tan 30° = 1/√3 = v² / (50 × 10) v² = (50 × 10) / √3 = 500 / 1.732 ≈ 288.7 v = √288.7 ≈ 17 m/s

Problem 4 A stone of mass 0.5 kg tied to a string of length 1 m is whirled in a vertical circle with speed 4 m/s at the lowest point. Find the tension at the lowest point and at the highest point. (g = 10 m/s²)

Solution At lowest point: T_low – mg = m v_low² / r T_low = m(g + v²/r) = 0.5(10 + 16/1) = 0.5(10 + 16) = 13 N

Speed at highest point (by conservation of energy): v_high² = v_low² – 4gr = 16 – 4×10×1 = 16 – 40 = –24 (impossible → stone will not reach the top; it will leave the circle earlier). (Note: Minimum v at bottom for full circle = √(5gr) = √50 ≈ 7.07 m/s. Here speed is too low.)Circular Motion Tool

Problem 5 A fan rotates at 1200 rpm. (a) Calculate angular velocity in rad/s. (b) If the tip of the blade is 0.2 m from the axis, find the linear speed of the tip.

Solution (a) 1200 rpm = 1200 revolutions per minute = 1200/60 = 20 rev/s ω = 2πf = 2π × 20 = 40π rad/s ≈ 125.66 rad/s

(b) v = rω = 0.2 × 40π = 8π m/s ≈ 25.13 m/s

Circular Motion Tool

इस टूल का उपयोग कैसे करें:Circular Motion Tool

1. प्रारंभिक बिंदु (Start Point): ट्रैक पर हरे रंग का बिंदु vᵢ (प्रारंभिक वेग) को दर्शाता है। डिफ़ॉल्ट रूप से, यह ऊपर की ओर (90°) इंगित करता है।Circular Motion Tool
2. एंगल स्लाइडर (Angle Slider): नीचे दिए गए स्लाइडर का उपयोग करके आप धावक (Runner) की अंतिम स्थिति तय कर सकते हैं। जैसे ही आप इसे हिलाते हैं, नीला चाप (Arc) तय की गई दूरी को दर्शाता है।
3. स्टार्ट सिमुलेशन (Start Simulation): इस बटन पर क्लिक करने से धावक शून्य से शुरू होकर आपके द्वारा चुने गए एंगल तक दौड़ता है। आप देख सकते हैं कि दौड़ते समय धावक की दिशा कैसे बदलती है।
4. एनालिटिक्स पैनल (Analytics Panel): दाईं ओर का पैनल रीयल-टाइम डेटा दिखाता है:
   • Δv (Velocity Change): यह वेक्टर अंतर को मापता है।Circular Motion Tool
   • a_avg (Average Acceleration): यह दिखाता है कि वेग परिवर्तन की दर क्या है।Circular Motion Tool
5. रीसेट (Reset): सिमुलेशन को उसकी मूल स्थिति में वापस लाने के लिए ‘Reset’ का उपयोग करें।Circular Motion Tool

इससे कौन लाभान्वित हो सकता है:

• छात्र (Class 9-12): जो छात्र भौतिकी (Physics) में काइनेमेटिक्स पढ़ रहे हैं, वे यह समझ सकते हैं कि गति (Speed) स्थिर होने के बावजूद त्वरण (Acceleration) क्यों होता है।
• शिक्षक (Physics Educators): कक्षा में ‘वेक्टर परिवर्तन’ और ‘दिशात्मक त्वरण’ जैसी अमूर्त अवधारणाओं को दृश्य रूप (Visualize) में समझाने के लिए यह एक बेहतरीन शिक्षण सहायक सामग्री है।
• प्रतियोगी परीक्षाओं के अभ्यर्थी (JEE/NEET): यह टूल वेक्टर सब्ट्रैक्शन और सर्कुलर मोशन के फॉर्मूलों के पीछे के तर्क को स्पष्ट करने में मदद करता है।Circular Motion Tool
• फिजिक्स के शौकीन: कोई भी व्यक्ति जो यह समझना चाहता है कि मोड़ पर मुड़ते समय बल और त्वरण कैसे काम करते हैं।

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