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
“Politics is the art of the possible”. Similarly, “Science is the art of the soluble”. Explain this beautiful aphorism on the nature and practice of science.
What are the dimensions of volume of a cube of edge a.
If all the terms in an equation have same units, is it necessary that they have same dimensions? If all the terms in an equation have same dimensions, is it necessary that they have same units?
If two quantities have same dimensions, do they represent same physical content?
Choose the correct statements(s):
(a) All quantities may be represented dimensionally in terms of the base quantities.
(b) A base quantity cannot be represented dimensionally in terms of the rest of the base quantities.
(c) The dimensions of a base quantity in other base quantities is always zero.
(d) The dimension of a derived quantity is never zero in any base quantity.
Find the dimensions of pressure.
Find the dimensions of electric field E.
The relevant equations 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.
Theory of relativity reveals that mass can be converted into energy. The energy E so obtained is proportional to certain powers of mass m and the speed c of light. Guess a relation among the quantities using the method of dimensions.
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.
Can you add three unit vectors to get a unit vector? Does your answer change if two unit vectors are along the coordinate axes?
Can a vector have zero component along a line and still have nonzero magnitude?
Is the vector sum of the unit vectors \[\vec{i}\] and \[\vec{i}\] a unit vector? If no, can you multiply this sum by a scalar number to get a unit vector?
Let \[\vec{A} = 3 \vec{i} + 4 \vec{j}\]. Write a vector \[\vec{B}\] such that \[\vec{A} \neq \vec{B}\], but A = B.
A vector is not changed if
A vector \[\vec{A}\] points vertically upward and \[\vec{B}\] points towards the north. The vector product \[\vec{A} \times \vec{B}\] is
Let A1 A2 A3 A4 A5 A6 A1 be a regular hexagon. Write the x-components of the vectors represented by the six sides taken in order. Use the fact the resultant of these six vectors is zero, to prove that
cos 0 + cos π/3 + cos 2π/3 + cos 3π/3 + cos 4π/3 + cos 5π/3 = 0.
Use the known cosine values to verify the result.
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.
Give an example for which \[\vec{A} \cdot \vec{B} = \vec{C} \cdot \vec{B} \text{ but } \vec{A} \neq \vec{C}\].
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.
The changes in a function y and the independent variable x are related as
\[\frac{dy}{dx} = x^2\] . Find y as a function of x.
