WBCHSE Class 11 Physics Notes For Different Kinds Of Motion

One-Dimensional Motion – Different Kinds Of Motion: Translation And Rotation

WBBSE Class 11 Types of Motion Notes

Particle: In practical cases, when a body is in motion it can rotate too. When a wheel is pushed, it moves forward. At the same time, it also rotates about an axis through its centre. A raindrop can vibrate while it falls.

• Representation of such motions are usually very complex. To avoid this complexity, an object is often taken as a geometrical point, ignoring its shape or size. This geometrical point is called a particle.
• In the case of linear motion, the properties of the particle and of the object are identical so discussion about the motion of the particle is sufficient to describe the motion of the object.
• To describe the motion of objects, sometimes we consider a body to be composed of many particles. In such cases, we do not consider a geometrical point but the aggregation of many particles.

Translation: If a body moves along a straight line, its motion is called translation. Motion of a freely falling body or the motion of a car along a straight road, are examples of translational motion.

Read and Learn More: Class 11 Physics Notes

Characteristics Of Translation:

1. The direction of motion remains the same.
2. The particles of an object under translatory motion traverse equal lengths in equal intervals of time and they also move parallel to one another. As in AA’A”, BB’B” and CC’C” lines are parallel and equal.
3. Lines joining any two particles of the body in translation remain parallel to one another for any position of the object. Observe, the lines AB, A’B’ and A”B” are parallel to one another.

The motion of any particle along a curved line can be considered as the aggregate of a number of infinitesimally small translatory motions.

Rotation: When an object moves in a circular path about a fixed point or an axis, the motion is called rotation. The axis is called the axis of rotation. Shows some examples of rotation. However, the axis may be located outside the object.

Linear Motion Explained for Class 11

Characteristics Of Rotation:

1. Each constituent particle of a rotating body rotates by an equal angle in a fixed interval of time.
2. The Axis of rotation always remains stationary.

Complex Motion: If a body exhibits translational as well as rotational motion simultaneously, then it is said to be in a state of complex motion.

Complex Motion Example:

1. The wheel of a running car executes a complex motion. The wheel rotates around the axis through its centre (rotation) and moves forward along the road (translation).
2. The earth rotates around its own axis and at the same time it revolves around the sun following an elliptical path. As the orbit is elliptical the Earth sometimes comes close to the sun and sometimes moves away from it. So, the earth also exhibits a complex motion.

Comparison Between Translation And Rotation:

1. Translation is motion in a straight line, whereas rotation is a circular motion in a plane.
2. In translation, the direction of motion is fixed. In rotation, the axis of rotation is fixed.
3. Translation of each constituent particle of the body is the same during the movement of the body. In rotation, constituent particles of the body at larger distances from the axis of rotation describe larger distances.

One-Dimensional Motion – Some Physical Quantities Related To Motion

Relative to a definite frame of reference, a body may be at rest, or in any form of motion like translation, rotation, vibration etc. For the convenience of the kinematical study of rest and motion of a body, a few important physical quantities are defined. These are the essential properties that represent the states of rest and motion of a body, and their measured values furnish the exact physical state.

Position: Let an object be located at the point A. To measure the position of the object, and express it in a well-defined manner, we have to

1. Choose a reference point, i.e., an origin—tiro point O is this origin.
2. Measure the linear distance between OA; and
3. Specify the direction of A relative to the origin O.

This would always lead to statements like: ‘The object A is situated 5 m east of the origin O’, or ‘The object B is 2 m north-east of O’. In short, we may write

⇒ \(\overrightarrow{O A}\) = 5 m towards east; \(\overrightarrow{O B}\) = 2 m to the north-east. These statements define the positions of the objects at A and at B. It is important to note that, each statement includes the magnitude of the linear distance, as well as the direction, relative to the origin.

Position Definition: The position of an object is defined as its linear distance as well as its direction with respect to a preassigned reference point.

Position Is A Vector Quantity: As per the definition, position is a physical quantity having both magnitude and direction. So it is a vector quantity. It is often called the position vector and denoted by the symbol \(\vec{r}\).

In the above examples, the position of A: \(\vec{r_1}\) = 5 in the east; the position of B: \(\vec{r_2}\) = 2 m north-east.

Another interesting point is that to find the position of B relative to A(\(\overrightarrow{A B}\)), a simple numerical calculation, using the values 5m and 2m, is not sufficient. The directions are to be considered as well. This technique leads to a new branch in mathematics, known as vector algebra.

Units And Dimension: The magnitude of the position vector is actually a distance. It has the units of length.

CCS system: cm

SI: m

Similarly, the dimension of the position vector is that of length, i.e., its dimension = L.

Displacement Definition: Displacement is defined as the change in position of a moving body in a fixed direction.

A and B are two fixed points. Many paths may exist between A and B. Three men move from A to B following different paths ACB, ADB and AEB. The lengths of these paths are different.

But as the initial and final positions of the men are the same, their displacements are also the same. The length of the minimum distance between A and B, i.e., the rectilinear path ADB is the measure of this displacement.

Magnitude And Direction Of Displacement: The length of the straight line connecting the initial and the final positions of a moving body is the magnitude of its displacement, and its direction is from the initial position to the final position along the straight line joining them. P and R are the initial and the final positions respectively of the body and the paths followed by it are PQ (3 m towards east) and QR (4 m towards north).

As defined, the displacement is  PR and it is independent of the path followed. From the measurements shown, PR =\(\sqrt{3^2+4^2}\) = 5 cm is the magnitude of displacement, and the direction is from P to R, shown by the arrowhead on PR.

Displacement Is A Vector Quantity: Displacement has both magnitude and direction and, hence, it is a vector quantity. It is represented by \(\overrightarrow{P R}\) in this case.

Zero Displacement: If a moving object starting from a point comes finally back to its initial position, then its displacement becomes zero.

Zero Displacement Example: A ball comes back to the hands of a thrower when it is thrown vertically upwards. The displacement of the ball is zero in this case. Hence, it can be concluded that the displacement of a moving object may be zero in spite of it travelling some distance.

Zero displacement is a null vector with magnitude zero and has no fixed direction.

Class 11 Physics	Class 12 Maths	Class 11 Chemistry
NEET Foundation	Class 12 Physics	NEET Physics

Geometric Representation Of Displacement: A reference frame helps to measure the magnitude and direction of a displacement. Let us Y consider a two-dimensional cartesian coordinate system where OX and OY are the two axes and O is the origin. Let a particle begin its journey from O and reach the point A(x, y).

The length OA gives the magnitude of the displacement of the particle— OA = \(\sqrt{x^2+y^2} \text {. }\). Now, if \(\overrightarrow{O A}\) makes an angle α with the X-axis, tanα = \(\frac{B A}{O B}\) = \(\frac{y}{x}\). In this case, we can say that the direction of displacement makes an angle α with the X-axis where α = tan^-1\(\frac{y}{x}\)

Oscillatory Motion Examples and Applications

For any particle in three-dimensional space, the displacement is represented by the straight line joining the initial and the final positions of the particle. If a particle travels from the point P₁(x₁, y₁,z₁) to the point P₂(x₂, y₂, z₂), then \(\overrightarrow{P_1 P_2}\) represents its displacement. The magnitude of the displacement is given by \(\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2} \text {. }\)

Unit And Dimension Of Displacement: The length of a straight line determines the magnitude of displacement. Hence, a unit of length is the unit of displacement and its dimension is the dimension of length i.e., L.

One-Dimensional Motion – Physical Quantities Related To Motion Numerical Examples

Short Answer Questions on Types of Motion

Example 1. A particle moves along a circular path of radius 7 cm. Estimate the distance covered and displacement when the particle

1. Covers half circular path and
2. Completes the total circular path once.

Solution:

Given

A particle moves along a circular path of radius 7 cm.

Circumference of the circular path = 2πr = 2 x \(\frac{22}{7}\) x 7 = 44 cm.

1. When it covers half the circumference, the particle moves from A to B along the path ACB. Hence, distance covered = \(\frac{44}{2}\) = 22 cm. Displacement is the length of straight line AB i.e., the diameter of the circle. Hence, displacement is 2 x 7 = 14 cm from A to B (\((\overrightarrow{A B})\)).
2. On completion of the total circular path ACB A, the distance covered is equal to the circumference of the circle = 44 cm. As the particle comes back to its initial position, displacement is zero.

Example 2. A particle moves 10√3 m towards the east and then 10 m towards the north. Find the magnitude and direction of its displacement.
Solution:

Given

A particle moves 10√3 m towards the east and then 10 m towards the north.

In this case AB = 10√3m, BC = 10m.

The initial and the final positions of the particle are A and C respectively.

∴ The magnitude of displacement,

A C = \(\sqrt{A B^2+B C^2}\)

= \(\sqrt{300+100}\)

= 20 m

If the angle between AC and AB is θ, then, \(\tan \theta=\frac{B C}{A B}=\frac{10}{10 \sqrt{3}}=\frac{1}{\sqrt{3}}\) or, θ = 30°.

This angle determines the direction of displacement.

Speed Definition: The distance travelled by a body in unit time is called its speed.

Distance is always measured along the path travelled by the moving body, irrespective of whether the path is straight or curved. Hence, if a body travels a length l in time t,

speed(v) = \(\frac{\text { distance travelled }(l)}{\text { time taken }(t)}\)

Speed is a scalar quantity.

Unit And Dimension Of Speed In Different Systems Of Units:

Unit of speed = \(\frac{\text { unit of length of the path travelled }}{\text { unit of time }}\)

CGS System: cm · s^-1

SI: m · s^-1

Dimension of speed = \(\frac{\text { dimension of distance }}{\text { dimension of time }}=\frac{\mathrm{L}}{\mathrm{T}}=\mathrm{LT}^{-1}\)

Relation Among Different Units: 1 m · s^-1 = 100 cm · s^-1

Dimension of speed = = k = LT^-1

In addition, km · h^-1 is also widely used.

1 km · h^-1 = \(\frac{1000}{60 \times 60} \mathrm{~m} \cdot \mathrm{s}^{-1}=\frac{5}{18} \mathrm{~m} \cdot \mathrm{s}^{-1} \text {. }\)

For easy recall, 18 km · h^-1 = 5 m · s^-1.

Average Speed: The speed of a body can be uniform or variable. When a body travels equal distances in equal intervals of time, its speed is uniform.

When distances travelled in equal intervals of time are unequal, the body moves with a variable speed.

For convenience, the average speed is often calculated in case of motion with variable speed. Dividing the total distance travelled by the total time taken to travel the distance, we get the average speed.

Thus if l₁, I₂, l₃ are the distances travelled by an object in times t₁, t₂ and t₃ respectively, then its average speed

= \(\frac{\text { total distance travelled }}{\text { total time taken to travel the distance }}=\frac{l_1+l_2+l_3}{t_1+t_2+t_3} \text {. }\)

The average speed is not an average of speeds.

Instantaneous Speed: The speed of a moving body at any instant is called its instantaneous speed.

Let us consider that a running (rain travels 10 in in 0,5 s. For the motion of a train, this 0.5 s lime Interval Is very small. So, this interval of time may be considered as an instant. Dividing the distance travelled by the train in that short interval of time gives the Instantaneous speed of the train. Hence, the instantaneous speed of the train = \(\frac{10}{0.65}\) = 20 m· s^-1.

Instantaneous Speed Definition: The instantaneous speed of a particle at a given point is the limiting value of the rate of the distance travelled with respect to a time when the time interval tends to zero.

Following the rule of differential calculus, if Δt is the time in which the distance travelled is Δl, then the instantaneous speed is

⇒ \(v_i=\lim _{\Delta t \rightarrow 0} \frac{\Delta l}{\Delta t}=\frac{d l}{d t}\)

where l is the distance or location of the particle along its locus from a given fixed point. For a body moving with uniform speed, the instantaneous speed at any instant is equal to the uniform speed.

For example, the speed of a moving car is measured by a speedometer. At any moment, the speedometer reads the instantaneous speed of the car. The pointer of the speedometer remains stationary when the car runs at a uniform speed. That is, the instantaneous speed is equal to the uniform speed of the car. The speedometer fluctuates when the car moves at varying speeds.

Distinction Between Average And Instantaneous Speed:

Average Speed: The total distance covered by a body in a certain interval of time, divided by the time interval is the average speed.

Instantaneous Speed: The rate of infinitesimal distance covered with respect to the corresponding infinitesimal time, is the instantaneous speed.

Velocity Definition: The rate of displacement of a body with time is called its velocity.

In other words, the rate of change of position of any object with respect to time is its velocity.

The change of position, i.e., the displacement is a vector quantity. If s Is the displacement of an object in time t, then,

velocity (v) = \(\frac{\text { displacement }(\mathrm{s})}{\text { time }(t)}\)

Velocity, like displacement, Is also a vector quantity.

Unit And Dimension Of Velocity: since the units of distance covered and of displacement are the same, the units of speed and velocity are also the same.

CGS System: cm · s^-1

SI: m · s^-1

Dimension of velocity = \(\frac{\text { dimension of displacement }}{\text { dimension of time }}\) = \(\frac{L}{T}\) = LT^-1

Therefore, the dimension of velocity is also identical to that of speed.

Uniform And Non-Uniform Velocity: if the velocity of a particle has a constant magnitude and a constant direction it is called uniform velocity. On the other hand, if the velocity of a particle changes with time, either in magnitude or in direction or in both, it is termed as a nonuniform velocity.

Due to gravity, the velocity of a falling body increases in magnitude keeping its direction unchanged. Therefore, the velocity of the body is non-uniform. Again, a car moving with a constant speed along a curved path has a non-uniform velocity due to its continuous change in direction.

A uniform circular motion is an example of a motion with uniform speed but non-uniform velocity.

Real-Life Applications of Linear, Rotary, and Oscillatory Motion

Average Velocity:

Average velocity, (v) = \(\frac{\text { total displacement }(s)}{\text { total time }(t)}\)

i. e., by dividing the total displacement of a particle in a certain interval of time by the time interval, its average velocity is obtained.

‘A stone takes 4 s to reach the ground when dropped from a height of 80 m ’—this statement provides no information about the change in velocity of the stone along the path. But it can be said that the average downward displacement of the stone per second is \(\frac{80}{4}\) or 20 m. So the average velocity of the stone is 20 m · s^-1.

Instantaneous Velocity: The velocity of a particle at any moment is called its instantaneous velocity. The instantaneous velocity can be defined similarly to the instantaneous speed.

Instantaneous Velocity Definition: The instantaneous velocity of a particle at a given point is the limiting value of the rate of 1 displacement from that point with respect to time when the time interval tends to zero.

Following the rule of differential calculus, if Δt is the time in which the displacement of any particle is Δs, then the instantaneous velocity is

⇒ \(v_i=\lim _{\Delta t \rightarrow 0} \frac{\Delta s}{\Delta t}=\frac{d s}{d t}\)

where s is the displacement of the particle from the given point.

Comparison Between Average Velocity And Instantaneous Velocity:

1. Average velocity = \(\frac{\text{total displacment}}{\text{total time}}\) But instantaneous velocity = \(\frac{ds}{dt}\). Indeed, the average velocity in an infinitesimally small interval of time is called the instantaneous velocity.
2. The instantaneous velocity becomes equal to the average velocity of a particle only if it moves with a uniform velocity. Otherwise, we cannot get any idea about the instantaneous velocities at different points from the average velocity of a particle.
3. In kinematics, ideas about the equality of velocities, the change in velocity, etc. are very important. The knowledge of the average velocity alone does not give any idea about them. Thus, the concept of instantaneous velocity is more important.

Comparison Between Speed And Velocity:

1. The rate of distance travelled with time is speed whereas the rate of displacement with time is velocity.
2. The units and dimensions of distance travelled and of displacement are the same. So the units and dimen¬sion of speed and those of velocity are the same.
3. Speed is a scalar quantity, but velocity is a vector quantity.
4. An object moving along a straight line with uniform speed has a uniform velocity as well, i.e., uniform velocity means a uniform speed in a fixed direction.
5. Uniform velocity always indicates uniform speed, but the converse is not true. A body moving with uniform speed in a curved path has a non-uniform velocity due to a change in its direction.
6. Speed is always positive or zero, but velocity may also be negative depending on the direction of motion.
7. The average speed of an object is zero means that the average velocity is zero too but the converse may not be true always.
8. Instantaneous speed and instantaneous velocity at any point of motion are the same in magnitude and independent of the shape of the path. But if an object follows a curved path, its average speed and average velocity at any interval of time are different in magnitude.