Introduction
Motion is a fundamental concept in physics that describes the change in position of an object over time. In the context of JEE Advanced Physics, understanding motion involves grasping several key principles and equations that govern how objects move. This study note will break down the topic of motion into manageable sections, covering everything from basic definitions to complex equations and applications.
Types of Motion
1. Translational Motion
Translational motion occurs when an object moves along a path in any of the three dimensions. This type of motion can be further divided into:
- Rectilinear Motion: Motion along a straight line.
- Curvilinear Motion: Motion along a curved path.
2. Rotational Motion
Rotational motion involves an object rotating about an axis. Each point in the object follows a circular path around the axis of rotation.
3. Oscillatory Motion
Oscillatory motion is repetitive back-and-forth movement about a central point, such as the motion of a pendulum.
Kinematics
Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion.
1. Displacement, Velocity, and Acceleration
- Displacement ($\vec{s}$): A vector quantity that denotes the change in position of an object. It has both magnitude and direction. $$\vec{s} = \vec{x}_f - \vec{x}_i$$ where $\vec{x}_f$ is the final position and $\vec{x}_i$ is the initial position.
- Velocity ($\vec{v}$): The rate of change of displacement with respect to time. It is also a vector quantity. $$\vec{v} = \frac{d\vec{s}}{dt}$$
- Acceleration ($\vec{a}$): The rate of change of velocity with respect to time. $$\vec{a} = \frac{d\vec{v}}{dt}$$
2. Equations of Motion for Uniform Acceleration
In cases of uniform acceleration, the following equations of motion can be used:
- $v = u + at$
- $s = ut + \frac{1}{2}at^2$
- $v^2 = u^2 + 2as$
where:
- $u$ is the initial velocity,
- $v$ is the final velocity,
- $a$ is the acceleration,
- $t$ is the time,
- $s$ is the displacement.
Remember to keep track of the direction when dealing with vector quantities. Use sign conventions consistently.
3. Projectile Motion
Projectile motion is a form of motion where an object moves in a bilaterally symmetrical, parabolic path under the influence of gravity alone.
Key Equations
- Horizontal Range: $R = \frac{u^2 \sin 2\theta}{g}$
- Maximum Height: $H = \frac{u^2 \sin^2 \theta}{2g}$
- Time of Flight: $T = \frac{2u \sin \theta}{g}$
A ball is thrown with an initial velocity of 20 m/s at an angle of 30° to the horizontal. Calculate the time of flight, maximum height, and horizontal range.
Solution
Given:
- $u = 20$ m/s
- $\theta = 30^\circ$
- $g = 9.8$ m/s²
Time of Flight: $$T = \frac{2u \sin \theta}{g} = \frac{2 \times 20 \times \sin 30^\circ}{9.8} = \frac{20}{9.8} \approx 2.04 \text{ s}$$
Maximum Height: $$H = \frac{u^2 \sin^2 \theta}{2g} = \frac{20^2 \times (\sin 30^\circ)^2}{2 \times 9.8} = \frac{400 \times 0.25}{19.6} \approx 5.10 \text{ m}$$
Horizontal Range: $$R = \frac{u^2 \sin 2\theta}{g} = \frac{20^2 \times \sin 60^\circ}{9.8} = \frac{400 \times \sqrt{3}/2}{9.8} \approx 35.36 \text{ m}$$
Dynamics
Dynamics is the study of the forces and torques that cause motion.
1. Newton's Laws of Motion
- First Law (Law of Inertia): An object will remain at rest or in uniform motion unless acted upon by a net external force.
Inertia is the resistance of any physical object to a change in its state of motion or rest.
- Second Law: The rate of change of momentum of an object is directly proportional to the net external force acting on it and occurs in the direction of the force. $$\vec{F} = m\vec{a}$$
- Third Law: For every action, there is an equal and opposite reaction.
2. Work, Energy, and Power
- Work: Work is done when a force causes displacement. $$W = \vec{F} \cdot \vec{s} = F s \cos \theta$$
- Kinetic Energy: The energy possessed by an object due to its motion. $$KE = \frac{1}{2}mv^2$$
- Potential Energy: The energy possessed by an object due to its position or configuration. $$PE = mgh$$
- Power: The rate at which work is done. $$P = \frac{dW}{dt} = \vec{F} \cdot \vec{v}$$
Circular Motion
1. Uniform Circular Motion
In uniform circular motion, an object moves in a circle with constant speed. Although the speed is constant, the velocity is not due to the continuous change in direction.
- Centripetal Acceleration: The acceleration directed towards the center of the circle. $$a_c = \frac{v^2}{r}$$
- Centripetal Force: The net force causing centripetal acceleration. $$F_c = m a_c = m \frac{v^2}{r}$$
2. Non-Uniform Circular Motion
In non-uniform circular motion, the speed of the object changes as it moves along the circular path. This involves both tangential acceleration ($a_t$) and centripetal acceleration ($a_c$).
Common MistakeDo not confuse centripetal force with centrifugal force. Centripetal force is the real force acting towards the center, while centrifugal force is a fictitious force observed in a rotating reference frame.
Relative Motion
Relative motion is the calculation of the motion of an object with respect to some other moving object.
1. Relative Velocity
If two objects A and B are moving with velocities $\vec{v}_A$ and $\vec{v}_B$ respectively, the velocity of A relative to B is given by:
$$\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$$
ExampleTwo cars are moving in the same direction with velocities 60 km/h and 80 km/h. Find the velocity of the first car relative to the second car.
Solution
Given:
- $\vec{v}_A = 60$ km/h
- $\vec{v}_B = 80$ km/h
Relative velocity: $$\vec{v}_{AB} = \vec{v}_A - \vec{v}_B = 60 - 80 = -20 \text{ km/h}$$
The negative sign indicates that the first car is moving slower than the second car by 20 km/h.
Conclusion
Understanding motion is crucial for solving problems in JEE Advanced Physics. By mastering the concepts of kinematics, dynamics, and relative motion, you will be well-equipped to tackle a wide range of questions. Remember to practice consistently and pay attention to the details, such as units and vector directions, to avoid common mistakes.