Mechanics: Circular Motion and Gravitation
What Is Circular Motion?
Circular motion occurs when an object moves in a circular path. It is governed by the principle that there is a constant force acting on the object to keep it in motion along the circle.
The force that keeps an object moving in a circle is called centripetal force. The magnitude of the centripetal force is given by the equation:
\[ F_c = \frac{mv^2}{r} \]
Where:
- \( F_c \) is the centripetal force (in newtons, N),
- \( m \) is the mass of the object (in kilograms, kg),
- \( v \) is the velocity of the object (in metres per second, m/s),
- \( r \) is the radius of the circular path (in metres, m).
Types of Circular Motion
Uniform Circular Motion
In uniform circular motion, the object moves around a circle at a constant speed, but its velocity is continuously changing due to the constant change in direction. The object’s acceleration is always directed towards the centre of the circle.
Non-Uniform Circular Motion
In non-uniform circular motion, the object’s speed is not constant. The object accelerates or decelerates as it moves along the circular path.
Gravitational Force and Newton’s Law of Universal Gravitation
Newton’s Law of Universal Gravitation
Newton’s Law of Universal Gravitation states that every particle of mass in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres. The formula is:
\[ F = \frac{Gm_1m_2}{r^2} \]
Where:
- \( F \) is the gravitational force (in newtons, N),
- \( G \) is the gravitational constant (\( 6.674 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \)),
- \( m_1 \) and \( m_2 \) are the masses of the two objects (in kilograms, kg),
- \( r \) is the distance between the centres of the two objects (in metres, m).
Applications of Circular Motion and Gravitation
Satellites and Orbital Motion
The motion of artificial satellites around Earth is governed by gravitational force. The centripetal force required to keep the satellite in orbit is provided by the force of gravity.
The velocity \( v \) of an object in orbit is given by:
\[ v = \sqrt{\frac{GM}{r}} \]
Where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth (or the central object),
- \( r \) is the radius of the orbit.
Tides and Gravitation
Tides are caused by the gravitational pull of the Moon and the Sun on Earth’s oceans. The varying gravitational force from the Moon causes the water to bulge, creating high and low tides.
Example Problem
Problem: A satellite orbits Earth at a height of \( 200 \, \text{km} \). Calculate the velocity of the satellite.
Solution:
Use the orbital velocity formula:
\[ v = \sqrt{\frac{GM}{r}} \]
Where:
- \( G = 6.674 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \),
- \( M = 5.97 \times 10^{24} \, \text{kg} \) (mass of Earth),
- \( r = 6.67 \times 10^6 + 200 \times 10^3 = 6.87 \times 10^6 \, \text{m} \) (radius of Earth plus height of satellite).
Substitute values into the equation:
\[ v = \sqrt{\frac{6.674 \times 10^{-11} \times 5.97 \times 10^{24}}{6.87 \times 10^6}} = \sqrt{\frac{3.98 \times 10^{14}}{6.87 \times 10^6}} = \sqrt{5.8 \times 10^7} = 7.6 \times 10^3 \, \text{m/s} \]
So, the velocity of the satellite is approximately \( 7.6 \times 10^3 \, \text{m/s} \).
Common Mistakes in Circular Motion and Gravitation
- Forgetting to Use the Correct Radius: When calculating centripetal force or orbital velocity, ensure you include the correct radius from the centre of the circular path.
- Not Accounting for Non-Uniform Motion: In non-uniform motion, additional forces like tangential acceleration need to be considered.
- Confusing Gravitational Force with Weight: Gravitational force is the force between two masses, while weight is the force exerted by Earth on an object.
Practice Questions
- Calculate the centripetal force acting on a car moving at \( 20 \, \text{m/s} \) around a curve with a radius of \( 50 \, \text{m} \).
- Using Newton’s Law of Gravitation, calculate the force between two objects with masses \( 5 \, \text{kg} \) and \( 10 \, \text{kg} \) placed \( 2 \, \text{m} \) apart.
- A satellite is in orbit at a height of \( 500 \, \text{km} \). Calculate the velocity of the satellite.