Physics Topics can also be used to explain the behavior of complex systems, such as the stock market or the dynamics of traffic flow.
What is an Example of Displacement?
Definition: Displacement is defined as the change in position of a moving body in a fixed direction.
In Fig., 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 R its direction is from the initial position to the final position along the straight line joining them.
In Fig., P and R are the initial and the final positions respectively of a 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 = [late]\sqrt{3^2+4^2}[/latex] = 5 m 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.
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.
Geometric representation of displacement: A reference frame helps to measure the magnitude and direction of a displacement. Let us consider a two dimensional cartesian coordinate system where OX and OY are the two axes and O is the origin [Fig.]. 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 = \(\sqrt{x^2+y^2}\). 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\(\frac{y}{x}\).
For any particle in three-dimènsional space, the displacement is represented by the straight line joining the initial and the final positions of the particle [Fig.]. 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 P1P2 = \(\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2}\).
Unit and dimension of displacement: The length of a straight line determines the magnitude of displacement. Hence, unit of length is the unit of displacement and its dimension is the dimension of length i.e., L.
Numerical Examples
Example 1.
A particle moves along a circular path of radius 7 cm. Estimate the distance covered and displacement when the particle
(i) covers half circular path and
(ii) completes the total circular path once.
Solution:
Circumference of the circular path
= 2πr = 2 × \(\frac{22}{7}\) × 7 = 44 cm.
i) When it covers half the circumference, the particle moves from A to B along the path ACB [Fig.].
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 × 7 = 14 cm from A to B (\(\overrightarrow{A B}\)).
ii) On completion of the total circular path ACBA, 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\(\sqrt{3}\) m towards east and then 10 m towards north. Find the magnitude and direction of its displacement.
Solution:
In this case AB = 10\(\sqrt{3}\)m, BC = 10 m [Fig.]
The initial and the final positions of the particle are A and C respectively.
∴ The magnitude of displacement,
AC = \(\sqrt{A B^2+B C^2}\)
= \(\sqrt{300+100}\)
= 20 m
If the angle between AC and AB is θ, then tanθ = \(\frac{B C}{A B}\) = \(\frac{10}{10 \sqrt{3}}\) = \(\frac{1}{\sqrt{3}}\) or, θ = 30°
This angle determines the direction of displacement.