How to Calculate Tension and Velocity at the Top and Bottom of a Vertical Circle
Motion in a vertical circle is a classic example of non-uniform circular motion in physics, where an object follows a circular path in a vertical plane under gravity. Understanding this is essential for JEE Main as it often appears in various forms, such as problems with a pendulum, a swing, or loops in roller coasters.
In motion in a vertical circle, both tension in the string (or normal reaction) and the force of gravity act on the object, and their directions and magnitudes change throughout the path. This combination leads to varying speed and tension, especially at the top and bottom of the circle.
Definition and Real Examples of Motion in a Vertical Circle
Motion in a vertical circle refers to the movement of an object along a circular path in a vertical plane, influenced by gravity. Real-life examples include:
- Children swinging on a swing
- Roller coaster loops
- Water flowing in a vertical pipe bend
- Stone tied to a string and whirled vertically
- Bucket of water swung in a vertical loop
Unlike horizontal circular motion, gravity plays a crucial role at all points here, affecting tension and speed asymmetrically. JEE Main often tests such conceptual contrasts.
Diagram and Force Analysis in Motion in a Vertical Circle
At any position on the vertical circle, two key forces act on the object:
- Weight (mg), always vertically downward
- Tension (T) in the string, always directed towards the centre
At the topmost point, both T and mg act downwards. At the bottom, T acts upwards, while mg is downwards. At the sides, their relative angles change, affecting net force and acceleration. Always sketch such diagrams for force clarity in the JEE exam.
Formulas and Stepwise Derivation for Motion in a Vertical Circle
To derive tension and velocity at key points, apply Newton’s laws and energy conservation. Let m be mass, R radius, v velocity, g gravity.
| Position | Tension Formula | Remarks |
|---|---|---|
| Bottom (B) | TB = m vB2/R + mg | Both T and mg upwards; T is highest |
| Top (T) | TT = m vT2/R - mg | Both T and mg downwards; T may be zero |
By conservation of mechanical energy:
- At bottom (B): E = ½ m vB2
- At top (T): E = ½ m vT2 + 2mgR
- Therefore, vB2 = vT2 + 4gR
For a complete revolution, tension at top should be ≥ 0:
m vT2/R - mg ≥ 0 ⇒ vT ≥ √ (gR).
So, the minimum speed at the lowest point for successful vertical circle is:
- vB,min = √(5gR)
Such step-by-step formula derivations are regularly tested in the JEE Main exam, especially within short answer numericals.
Work-Energy Analysis for Different Positions
To determine velocities at the top, sides, or bottom, use the work-energy theorem. Mechanical energy is conserved (if no air resistance):
| Position | Kinetic Energy | Potential Energy |
|---|---|---|
| Bottom (B) | ½ m vB2 | 0 (reference) |
| Top (T) | ½ m vT2 | 2mgR |
| Any height h | ½ m v2 | mgh |
Work-energy method ensures systematic calculation of velocities at all points. This connects strongly with energy topics in work, energy and power chapters.
Key Differences: Motion in Vertical and Horizontal Circles
| Vertical Circle | Horizontal Circle |
|---|---|
| Gravity always acts, affecting speed | Gravity acts perpendicular to the motion |
| Tension (or normal) changes with position | Tension (or normal) usually constant |
| Non-uniform speed | Speed can be uniform |
Students often confuse non-uniform circular motion in the vertical plane with the uniform case. Highlight these differences for JEE MCQs.
Common Examples and Applications in Motion in a Vertical Circle
Applications of vertical circles are seen in:
- Pendulums and swings in playgrounds
- Roller coaster loops and thrill rides
- Water in buckets or pipes moving in a loop
- Satellites (for analytical extensions)
- High-speed road curves with banking
Each real context demands careful force analysis, especially for JEE Main application questions.
Solving Numericals: JEE Main Practice Example on Motion in a Vertical Circle
Example: A 0.5 kg mass tied to a 1 m long string moves in a vertical circle. What is the minimum speed at the lowest point to just complete the circle? (g = 10 m/s2)
- Minimum speed formula: vB,min = √(5gR)
- Substitute values: vB,min = √(5 × 10 × 1) = √50 m/s
- Calculate: vB,min ≈ 7.07 m/s
This method matches JEE Main-level requirements. For advanced practice, see the topic’s dedicated page and kinematics mock test.
Practical Tips and Links for Mastering Motion in a Vertical Circle
- Draw neat force diagrams at each key point
- Always use energy conservation from bottom to top
- Remember tension is minimum at the top, maximum at the bottom
- Don’t forget units in SI (kg, m, s)
- Review related topics: centripetal force, laws of motion, and rotational motion
Explore further with circular motion, gravitation, and work, energy and power revision notes for cross-topic mastery.
For more concise guides, formula tables, and expert strategies on JEE Main Physics concepts like motion in a vertical circle, trust Vedantu’s experienced educators and well-illustrated resources tailored for exam success.
FAQs on Motion in a Vertical Circle Explained with Diagrams, Derivations & Problems
1. What is motion in a vertical circle?
Motion in a vertical circle refers to the movement of an object along a circular path oriented vertically, under the influence of gravity and tension. Key points include:
- The object is usually tied to a string or moves along a rigid path in a vertical plane.
- Both tension in the string and gravitational force act on the object, with their directions and magnitudes changing with position (top, bottom, sides).
- Examples include a swinging pendulum, roller coasters, and bucket swings.
2. What forces act on a body in vertical circular motion?
In vertical circular motion, two main forces act on the object:
- Gravity (weight, mg) acts vertically downward at all positions.
- Tension (T) in the string or normal force from the path acts towards the center of the circle.
3. How do you calculate tension at the top and bottom of a vertical circle?
The tension at the top and bottom of a vertical circle can be found using force balance and centripetal force equations:
- At bottom (TB): TB = m vB2/r + mg
- At top (TT): TT = m vT2/r − mg
4. What is the minimum velocity required at the top of a vertical circle for complete revolution?
The minimum velocity at the top ensures that tension in the string just remains non-zero, keeping the object in circular motion. It is given by:
- vmin, top = sqrt(g r)
5. How is vertical circular motion different from horizontal circular motion?
Vertical circular motion involves changes in speed and force directions due to gravity, while horizontal circular motion typically has constant speed. Main differences:
- Gravity affects vertical circle speed and tension/normal force throughout the path.
- In the horizontal circle (like a rotating table), gravity acts perpendicular and does not affect the centripetal force directly.
- Vertical motion requires energy analysis (work-energy principles) for different positions.
6. Which real-life examples use vertical circular motion?
Real-life applications of vertical circular motion include:
- Roller coaster loops
- Swings and pendulums
- Water bucket spun in a circle
- Terminating pipe flows (liquid motion)
- Amusement park rides
7. How does energy conservation apply in vertical circular motion?
In vertical circular motion, total mechanical energy (kinetic + potential) is conserved in the absence of air resistance:
- At different points, kinetic energy and gravitational potential energy convert into each other.
- Work-energy principle helps calculate velocities and tensions at various positions in the circle.
- For a complete revolution, the energy at the bottom must be sufficient to reach the top with the minimum required velocity.
8. What happens if the velocity is less than the minimum value at the top of the circle?
If the velocity at the top falls below sqrt(g r), the tension becomes zero and the object cannot maintain the circular path:
- The object will fail to complete the circle and may fall due to gravity.
- Maintaining minimum velocity at the top ensures continuous circular motion.
9. Why is the tension minimum at the top of a vertical circle?
The tension in the string is minimum at the top of the vertical circle because gravity and tension both act towards the center, reducing the tension required to provide centripetal force. At the bottom, tension must counteract gravity as well, so it is maximum there.
10. How to solve numericals based on motion in a vertical circle?
To solve vertical circular motion problems:
- Draw a labeled diagram showing forces at key points (top, bottom, sides).
- Use energy conservation to find velocities at different positions.
- Apply tension formulas using Newton’s Laws and centripetal force concepts.
- Substitute known values and solve algebraically for the required result.
