Units and Dimensions

A vector of magnitude 10 m in the direction $37^{\circ}$ south of west has its initial point at $(5\ m, 2\ m)$ . If positive x-axis represents the east and positive y-axis the north, the coordinates of its terminal point are

After firing, a bullet is found to move at an angle of $37^{\circ}$ to horizontal. Its acceleration is $10 \, m/s^{2}$ downwards. Find the component of acceleration in the direction of the velocity.

A string connected with bob is suspended by the point $(O)$ such that it sweeps out conical surface in horizontal plane. Here $\vec{r}$ is the position vector of bob, $\vec{v}$ is its velocity and $\vec{z}$ is the axis of swept cone as shown. Select INCORRECT statement :-

Two forces P and Q of magnitude 2f and 3f, respectively, are at an angle $\theta$ with each other. If the force Q is doubled, then their resultant also gets doubled. Then, the angle is:

Consider three vectors $\vec{A} = \hat{i} + \hat{j} - 2\hat{k}, \vec{B} = \hat{i} - \hat{j} + \hat{k}$ and $\vec{C} = 2\hat{i} - 3\hat{j} + 4\hat{k}$ . A vector $\vec{X}$ of the form $\alpha\vec{A} + \beta\vec{B}$ ( $\alpha$ and $\beta$ are numbers) is perpendicular to $\vec{C}$ . The ratio of $\alpha$ and $\beta$ is

Given the vectors $\vec{A} = 2\hat{i} + 3\hat{j} - \hat{k}$ ; $\vec{B} = 3\hat{i} - 2\hat{j} - 2\hat{k}$ and $\vec{C} = p\hat{i} + p\hat{j} + 2p\hat{k}$ . Find the angle between $(\vec{A} - \vec{B})$ and $\vec{C}$

Two forces $(\hat{i} + \hat{j} + \hat{k})N$ and $(\hat{i} + 2\hat{j} + 3\hat{k})N$ act on a particle and displace it from (2, 3, 4) to point (5, 4, 3). Displacement is in m. Work done is:

If mass is expressed as $v^{x} d^{y} a^{z}$ where v is velocity; d is density and a is acceleration then the value of $x + y + z$ is

The angle subtended by the moon's diameter at a point on the earth is about $0.50^{\circ}$ . Use this and the fact that the moon is about 384000 km away to find the approximate diameter of the moon (in km).

Three particles P, Q and R are moving along the vectors $\vec{A} = \hat{i} + \hat{j}, \vec{B} = \hat{j} + \hat{k}$ and $\vec{C} = -\hat{i} + \hat{j}$ respectively. They strike on a point and start to move in different directions. Now particle P is moving normal to the plane which contains vector $\vec{A}$ and $\vec{B}$ . Similarly, particle Q is moving normal to the plane which contains vector $\vec{A}$ and $\vec{C}$ . The angle between the direction of motion of P and Q is $\cos^{-1}\left(\frac{1}{\sqrt{x}}\right)$ . Then the value of x is ____.

During a war between Ra1 and G1, the power shot should be by G1 reaches to Ra1. It was found that the power of the power shot fired depends on mass $(m_{0})$ of G1, velocity $(v_{0})$ of his hand and time lag $(t_{0})$ between his thought of firing & actual time when shot was fired. If power of power shot depends on $k^{th}$ power of velocity of his hand, fill k in OMR sheet.

If the resultant of two forces of magnitudes P and Q acting at a point at an angle of $60^{\circ}$ is $\sqrt{7}$ Q, then P/Q is

Two forces $\vec{F}_1$ and $\vec{F}_2$ of magnitude $10\sqrt{10} N$ each are inclined at an angle of $1.8^{\circ}$ to each other. What is the magnitude (in $N$ ) of vector $\vec{F}_1 - \vec{F}_2$ ? (Take $\pi^2 = 10$ .)

If $\hat{i} + 2\hat{j} - n\hat{k}$ is perpendicular to $4\hat{i} + 2\hat{j} + 2\hat{k}$ , then the value of n is

A force $\vec{F}=5\hat{i}+2\hat{j}+\hat{k}$ displaces a body from a point of coordinate $(1,1,1)$ to another point of coordinates $(2,0,3)$ . Calculate the work done (in J) by the force.

Three vectors $\vec{P},\vec{Q}$ and $\vec{R}$ are such that $|\vec{P}| = |\vec{Q} |,\left|\vec{R}\right| = \sqrt{2} |\vec{P} |$ and $\vec{P} +\vec{Q} +\vec{R} = \vec{0}$ . If the angle between $\vec{P}$ and $\vec{R}$ is $\frac{a\pi}{4}$ (in radians) then find the value of $(\alpha)$ .

The sum of two force acting at a point is 16N. If their resultant is normal to the smaller force and has a magnitude of 8N, then find the value of smaller force.

If momentum (P), area (A) and time (T) are taken to be the fundamental quantities then the dimensional formula for energy is:

A boy A is standing $20\sqrt{3}$ away in a direction $30^\circ$ north of east from his friend B. Another boy C standing somewhere east of B can reach A, if he walks in a direction $60^\circ$ north of east. In a Cartesian coordinate system with its x-axis towards the east, the position of C with respect to A is

A body moves in anticlockwise direction on a circular path in the $x$-$y$ plane. The radius of the circular path is $5,m$ and its centre is at the origin. At an interval of time, displacement of the body is observed to be $6,m$ in the positive $y$-direction. Which of the following is true?

A particle moves from a position $3\hat{i}+2\hat{j}-6\hat{k}$ to a position $14\hat{i}+13\hat{j}+9\hat{k}$ in $m$ and a uniform force of $4\hat{i}+3\hat{k},N$ acts on it. The work done by the force is :-

The vector $\hat{i} + x\hat{j} + 3\hat{k}$ is rotated through an angle $\theta$ and doubled in magnitude, then it becomes $4\hat{i} + (4x - 2)\hat{j} + 2\hat{k}$ . The values of $x$ are

The value of $\left|\vec{A}+\vec{B}-\vec{C}+\vec{D}\right|$ can be zero if:-

Priya says that the sum of two vectors by the parallelogram method is $\vec{R} = 5\hat{i}$ . Subhangi says it is $\vec{R} = \hat{i} + 4\hat{j}$ . Both used the parallelogram method, but one used the wrong diagonal. Which one of the vector pairs below contains the original two vectors?

PARAGRAPH FOR QUESTION NO. 10 TO 12 A physical quantity is a physical property of a phenomenon, body, or substance, that can be quantified by measurement. The magnitude of the components of a vector are to be considered dimensionally distinct. For example, rather than an undifferentiated length unit L, may represent length in the x direction as $L_{x}$ , and so forth. This requirement stems ultimately from the requirement that each component of a physically meaningful equation (scalar or vector) must be dimensionally consistent. As an example, suppose wish to calculate the drift S of a swimmer crossing a river flowing with velocity $V_{x}$ and of width D and he is swimming in direction perpendicular to the river flow with velocity $V_{y}$ relative to river, assuming no use of directed lengths, the quantities of interest are then $V_{x}, V_{y}$ both dimensioned as $\frac{L}{T}$ , S the drift and D width of river both having dimension L. With these four quantities, may conclude that the equation for the drift S may be written: $S = V_{x}^{a} V_{y}^{b} D^{c}$ Or dimensionally $L = \left(\frac{L}{T}\right)^{a + b} \times (L)^{c}$ from which may deduce that $a + b + c = 1$ and $a + b = 0$ , which leaves one of these exponents undetermined. If, however, use directed length dimensions, then $V_{x}$ will be dimensioned as $\frac{L_{x}}{T}$ , $V_{y}$ as $\frac{L_{y}}{T}$ , S as $L_{x}$ and D as $L_{y}$ . The dimensional equation becomes: $L_{x} = \left(\frac{L_{x}}{T}\right)^{a} \left(\frac{L_{y}}{T}\right)^{b} \left(L_{y}\right)^{c}$ and may solve completely as a = 1, b = -1 and c = 1. The increase in deductive power gained by the use of directed length dimensions is apparent. Which of the following is not a physical quantity