Advertisements
Advertisements
Question
Refer to figure (2 − E1). Find (a) the magnitude, (b) x and y component and (c) the angle with the X-axis of the resultant of \[\overrightarrow{OA}, \overrightarrow{BC} \text { and } \overrightarrow{DE}\].
Advertisements
Solution
First, let us find the components of the vectors along the x and y-axes. Then we will find the resultant x and y-components.
x-component of \[\overrightarrow{OA} = 2 \cos 30^\circ= \sqrt{3}\]
x-component of \[\overrightarrow{BC}\] = 1.5 cos120°
\[= - \frac{\left( 1 . 5 \right)}{2} = - 7 . 5\]
x-component of \[\overrightarrow{DE}\] = 1 cos 270°
= 1 × 0 = 0 m
y-component of \[\overrightarrow {OA}\] = 2 sin 30° = 1
y-component of \[\overrightarrow{BC}\] = 1.5 sin 120°
\[= \frac{\left( \sqrt{3} \times 1 . 5 \right)}{2} = 1 . 3\]
y-component of \[\overrightarrow{DE}\] = 1 sin 270° = −1
x-component of resultant \[R_x = \sqrt{3} - 0 . 75 + 0 = 0 . 98 m\]
y-component of resultant Ry = 1 + 1.3 − 1 = 1.3 m
\[\therefore \text { Resultant, R }= \sqrt{\left( R_x \right)^2 + \left( R_y \right)^2}\]
\[ = \sqrt{\left( 0 . 98 \right)^2 + \left( 1 . 3 \right)^2}\]
\[ = \sqrt{0 . 96 + 1 . 69}\]
\[ = \sqrt{2 . 65}\]
\[ = 1 . 6 m\]
If it makes an angle α with the positive x-axis, then
\[\tan \alpha = \frac{\text { y -component }}{\text { x - component }}\]
\[ = \frac{1 . 3}{0 . 98} = 1 . 332\]
APPEARS IN
RELATED QUESTIONS
If two quantities have same dimensions, do they represent same physical content?
It is desirable that the standards of units be easily available, invariable, indestructible and easily reproducible. If we use foot of a person as a standard unit of length, which of the above features are present and which are not?
Choose the correct statements(s):
Find the dimensions of frequency .
Find the dimensions of magnetic field B.
The relevant equation are \[F = qE, F = qvB, \text{ and }B = \frac{\mu_0 I}{2 \pi a};\]
where F is force, q is charge, v is speed, I is current, and a is distance.
Test if the following equation is dimensionally correct:
\[h = \frac{2S cos\theta}{\text{ prg }},\]
where h = height, S = surface tension, ρ = density, I = moment of interia.
Test if the following equation is dimensionally correct:
\[V = \frac{\pi P r^4 t}{8 \eta l}\]
where v = frequency, P = pressure, η = coefficient of viscosity.
Test if the following equation is dimensionally correct:
\[v = \frac{1}{2 \pi}\sqrt{\frac{mgl}{I}};\]
where h = height, S = surface tension, \[\rho\] = density, P = pressure, V = volume, \[\eta =\] coefficient of viscosity, v = frequency and I = moment of interia.
Let x and a stand for distance. Is
\[\int\frac{dx}{\sqrt{a^2 - x^2}} = \frac{1}{a} \sin^{- 1} \frac{a}{x}\] dimensionally correct?
Is it possible to add two vectors of unequal magnitudes and get zero? Is it possible to add three vectors of equal magnitudes and get zero?
The x-component of the resultant of several vectors
(a) is equal to the sum of the x-components of the vectors of the vectors
(b) may be smaller than the sum of the magnitudes of the vectors
(c) may be greater than the sum of the magnitudes of the vectors
(d) may be equal to the sum of the magnitudes of the vectors.
Add vectors \[\vec{A} , \vec{B} \text { and } \vec{C}\] each having magnitude of 100 unit and inclined to the X-axis at angles 45°, 135° and 315° respectively.
A spy report about a suspected car reads as follows. "The car moved 2.00 km towards east, made a perpendicular left turn, ran for 500 m, made a perpendicular right turn, ran for 4.00 km and stopped". Find the displacement of the car.
Suppose \[\vec{a}\] is a vector of magnitude 4.5 units due north. What is the vector (a) \[3 \vec{a}\], (b) \[- 4 \vec{a}\] ?
Let \[\vec{a} = 2 \vec{i} + 3 \vec{j} + 4 \vec{k} \text { and } \vec{b} = 3 \vec{i} + 4 \vec{j} + 5 \vec{k}\] Find the angle between them.
Prove that \[\vec{A} . \left( \vec{A} \times \vec{B} \right) = 0\].
The electric current in a charging R−C circuit is given by i = i0 e−t/RC where i0, R and C are constant parameters of the circuit and t is time. Find the rate of change of current at (a) t = 0, (b) t = RC, (c) t = 10 RC.
Round the following numbers to 2 significant digits.
(a) 3472, (b) 84.16. (c)2.55 and (d) 28.5
In a submarine equipped with sonar, the time delay between the generation of a pulse and its echo after reflection from an enemy submarine is observed to be 80 s. If the speed of sound in water is 1460 ms-1. What is the distance of an enemy submarine?
Jupiter is at a distance of 824.7 million km from the Earth. Its angular diameter is measured to be 35.72˝. Calculate the diameter of Jupiter.
