How to Derive the Formula and Tension in a Conical Pendulum?
Conical pendulum is a classic example of uniform circular motion in Physics, highly relevant for JEE Main. Unlike a simple pendulum, the bob in a conical pendulum traces a horizontal circle, while the string sweeps out the surface of a cone. This setup illustrates tension and centripetal force concepts, which are tested frequently in competitive exams.
In a typical JEE Main problem on conical pendulum, you’re expected to analyze the forces, draw the free body diagram, derive the formulae, and apply the concepts to calculation tasks. Mastering this topic strengthens your understanding of circular motion, banking of roads, and practical engineering mechanisms.
What is a Conical Pendulum?
A conical pendulum consists of a small bob of mass m tied to a light, inextensible string of length l. The string is fixed at its upper end. When set in motion, the bob moves in a horizontal circle of radius r at constant speed. The string makes a constant angle θ with the vertical, forming a cone.
This arrangement demonstrates uniform circular motion and introduces essential ideas like centripetal force and angular velocity. Unlike vertical circular motion, here the path’s plane remains horizontal. For further clarity on related circular motion fundamentals, see circular motion and centripetal force pages.
Conical Pendulum Diagram and Force Analysis
Always start by drawing a labelled diagram for the conical pendulum: show the fixed point, the angled string, the horizontal circle, and key parameters like θ, l, and r. Next, sketch the free body diagram at the bob.
- Weight (mg) acts vertically downwards
- Tension (T) acts along the string
- Centripetal force is provided by the horizontal component of tension
- The string makes an angle θ with the vertical
Setting up this force diagram is a crucial step in almost every JEE Main conical pendulum problem. For more on force decomposition, visit laws of motion and tension normal and spring forces.
Conical Pendulum Formulae and Derivation
Calculate the radius of the horizontal circle as r = l sin θ. The components of tension are critical for the balance:
- Vertical: T cos θ = mg
- Horizontal: T sin θ = m v2/r
Using these, the angular velocity ω and the period (Tp) of a conical pendulum can be expressed as:
| Formula | Meaning | Variables |
|---|---|---|
| r = l sin θ | Radius of horizontal circle | l: string length, θ: angle |
| v = \(\sqrt{g l \sin^2\theta / \cos\theta}\) | Speed of the bob | g: gravity, l, θ |
| T_p = 2\pi\sqrt{\frac{l \cos\theta}{g}} | Time period of revolution | T_p: time, l, θ, g |
| T = \frac{mg}{\cos\theta} | Total string tension | m: mass, g, θ |
When solving numericals, always express answers in SI units and check angle conventions. Key pitfalls: confusing tension direction and forgetting to resolve forces properly.
Conical Pendulum vs Simple Pendulum (JEE Perspective)
| Aspect | Conical Pendulum | Simple Pendulum |
|---|---|---|
| Path | Horizontal circle | Vertical arc |
| Forces | Tension & gravity; tension has vertical and horizontal components | Tension & gravity; tension only vertical |
| Period depends on | Length and angle | Only length |
| Application | Centrifugal devices, amusement rides | Clocks, timekeeping |
Always contrast these in conceptual answers. For detailed coverage of the simple pendulum, refer to simple pendulum. To understand differences across other motions, see difference between circular and rotational motion and difference between uniform circular motion and non uniform circular motion.
Solving a Typical Conical Pendulum Question
Suppose a bob of mass 0.2 kg is suspended by a 1 m string and rotates such that the string makes a 30° angle with the vertical. Find the tension in the string.
- Resolve forces: T cos 30° = mg
- Calculate T = mg / cos 30°
- Substitute: 0.2 × 9.8 / 0.866
- Get T = 2.26 N
In JEE problems, remember you must check angle use and ensure all trigonometric functions are correct for SI units. For more practice, review laws of motion practice paper and work energy and power practice paper.
Practical Applications and Common Mistakes in Conical Pendulum
- Used in centrifuges and mixer-grinders
- Describes motion of brake drums and fairground rides
- Linked to banking of roads and curves
- Crucial in certain laboratory clocks and old timekeeping devices
- Forgetting to use angle with the vertical
- Not writing all units in SI system
- Confusing radius with string length
- Ignoring horizontal force balance
JEE Main often tests if you can spot such errors. For similar mechanical links, see rotational motion and motion in 2d dimensions pages.
This topic also helps when studying gravitation, banking curves (if present), and work energy and power.
Vedantu’s JEE Physics resources clarify conical pendulum nuances and link with many practice questions. For revision, you may consult physics revision notes and kinematics important questions as well.
Mastering the concepts, diagrams, and standard formulae of conical pendulum ensures confidence in JEE Main numerical, theoretical, and application sections.
FAQs on Conical Pendulum – Definition, Working, and Application
1. What is a conical pendulum?
A conical pendulum is a mass attached to a string or rod that moves in a horizontal circular path, causing the string to trace out a cone shape. In this setup, the mass rotates uniformly in a horizontal circle while the string consistently makes an angle with the vertical.
Key points regarding conical pendulum are:
- Demonstrates uniform circular motion in a horizontal plane
- The string makes a constant angle with the vertical, tracing a cone
- Tension in the string provides the needed centripetal force
- Commonly asked in JEE, NEET, and Board Physics exams
2. How does a conical pendulum work?
A conical pendulum works by rotating a suspended mass in a horizontal circle, where the string traces a cone. The tension and gravity balance to maintain this motion.
Working principles:
- The mass moves with constant speed in a horizontal circle
- The tension in the string has both horizontal (centripetal) and vertical (weight balancing) components
- The angle of the string stays constant due to uniform motion
- Circular motion formulas are used to calculate tension, period, and radius
3. How do you derive the formula for the period of a conical pendulum?
The period (T) of a conical pendulum can be derived using the balance of forces and circular motion principles.
Derivation steps:
- Vertical component of tension balances the weight: T cos θ = mg
- Horizontal component gives centripetal force: T sin θ = mv²/r
- Relate radius (r = l sin θ) and solve for period: T = 2π √(l cos θ / g)
- The angle θ, string length l, and g (acceleration due to gravity) affect the period
4. What is the difference between a conical pendulum and a simple pendulum?
A conical pendulum traces a horizontal circle while a simple pendulum swings back and forth in a plane.
Main differences:
- Path: Conical – horizontal circle; Simple – arc (plane)
- Motion: Conical – uniform circular; Simple – oscillatory
- Angle: Conical – fixed angle with vertical; Simple – angle changes with swing
- Formula for period: Conical involves angle θ; Simple: T = 2π√(l/g)
- Exam use: Both appear in JEE, NEET, conceptual and formula-based questions
5. What is the formula for tension in a conical pendulum?
The tension (T) in a conical pendulum string is given by:
- T = mg / cos θ, where θ is the angle with vertical, m is mass, g is acceleration due to gravity
- // T has both vertical (balances weight) and horizontal (provides centripetal force) components
- The formula is key for solving circular motion and JEE Main Physics problems on conical pendulums
6. What are real-life applications of a conical pendulum?
The conical pendulum concept is used in various real-life applications involving uniform circular motion. Examples include:
- Centrifugal governors in steam engines
- Amusement park rides (swing rides)
- Aeroplane turns and banking of roads
- Design of rotating laboratory instruments
7. What mistakes do students make when solving conical pendulum problems in exams?
Common mistakes in conical pendulum problems often lead to lost marks in JEE, NEET, and other exams. Typical errors include:
- Confusing the tension formula with that of a simple pendulum
- Incorrectly drawing the free body diagram
- Using wrong components for centripetal force and weight
- Missing out on the angle θ in the formulae
- Mixing up period formulas of simple and conical pendulums
8. How do you draw the free body diagram of a conical pendulum?
To draw a free body diagram (FBD) of a conical pendulum:
- Show the mass (m) at the end of the string
- Draw two forces: tension (T) along the string, and weight (mg) vertically down
- Resolve T into vertical (T cos θ) and horizontal (T sin θ) components
- Indicate angle θ between string and vertical line
9. Can a conical pendulum be used as a clock like a simple pendulum?
While a simple pendulum is traditionally used in clocks due to its predictable period, a conical pendulum can also be used in timing devices under certain conditions. However, its period depends on the angle, making it less common for accurate timekeeping.
- Simple pendulum clocks are more widespread because their period is independent of amplitude for small angles
- Conical pendulum clocks require precise control of angle θ to keep accurate time
10. What is the centripetal force in a conical pendulum and how is it produced?
The centripetal force in a conical pendulum is provided by the horizontal component of tension in the string, keeping the mass in a uniform circular path.
- Fc = T sin θ, where T is tension, θ is angle with vertical
- This force always points toward the center of the circle, enabling circular motion
- It is different from the vertical component, which balances the weight
