CDF POINTS
1.The scalar product of the two vectors, \(
\overrightarrow {P\text{ }} and\text{ }\overrightarrow Q
\) is defined as \(
\overrightarrow P .\overrightarrow Q = \left| {\overrightarrow P } \right|\left| {\overrightarrow Q } \right|\cos \theta \). Here \(\theta\) is the angle between \(
\overrightarrow {P\text{ }} and\text{ }\overrightarrow Q \).
The angle between the two vectors \(\overrightarrow {P\text{ }} and\text{ }\overrightarrow Q \) is given by \(
\cos \theta = \frac{{\overrightarrow P .\overrightarrow Q }}
{{\left| {\overrightarrow P } \right|\left| {\overrightarrow Q } \right|}}\)
2.It is always a scalar which is positive if angle between the vectors is acute (i.e., <900) and negative if angle between them is obtuse (i.e., \(
90^0 < \theta \leqslant 180^0 \))
3.Scalar product of two vectors will be maximum when , vectors are parallel.
\(
\Rightarrow \left( {\overrightarrow P .\overrightarrow Q } \right)_{\max } = PQ
\)
4.In case of unit vector \(\hat n\),
\(
\hat n .\hat n = 1 \times 1 \times \cos \theta = 1\)
\(
\hat n .\hat n = \hat i .\hat i = \hat j .\hat j = \hat k.\hat k = 1
\)
In case of orthogonal unit vectors \(\hat i.\hat j\) and \(
\hat k .\hat i.\hat j = \hat j .\hat k = \hat k .\hat i = 0
\)
5.If \(\overrightarrow A\) and \(\overrightarrow B\) are two vectors and the angle between them is then the cross product of these two vectors is given by \(
\overrightarrow A \times \overrightarrow B = \left( {AB\sin \theta } \right)\hat n
\).
Where \(\hat n\) is unit vector perpendicular to the plane containing \(
\overrightarrow A \,\,\& \,\overrightarrow B
\).
6.Unit vector normal to both \(\overrightarrow A\) and \(\overrightarrow B\) is \(
\hat n = \frac{{\overrightarrow A \times \overrightarrow B }}
{{\left| {\overrightarrow A \times \overrightarrow B } \right|}}
\).
7.If two vectors are parallel (\(\theta =0\)) or anti parallel (\(\theta=180^0\) ) then \(\overrightarrow A \times \overrightarrow B = 0
\)
8.If two vectors \(
\overrightarrow A\) and \(\overrightarrow B\) are parallel \(
\frac{{A_x }}
{{B_X }} = \frac{{A_y }}
{{B_y }} = \frac{{A_z }}
{{B_z }} = \text{constant}
\) or \(
\overrightarrow A \times \overrightarrow B = 0
\)
9.If two vectors are perpendicular to each other \(
\theta = 90^0
\) then \(\left| {\overrightarrow A \times \overrightarrow B } \right| = AB
\) (maximum)
10.When a particle completes one revolution the angular displacement is \(
\theta = 2\pi
\) radian
11.Average angular velocity \(
\omega _{avg} = \frac{{\Delta \theta }}
{{\Delta t}}
\)
instantaneous angular velocity is
\(
\omega = \mathop {\lim }\limits_{\Delta t \to 0} \left[ {\frac{{\Delta \theta }}
{{\Delta t}}} \right] = \frac{{d\theta }}
{{dt}}
\)
SI unit rad S-1
* Dimensional formula [ T-1]
12.The instantaneous angular velocity at the instant of time ‘t’ is
\(
\omega = \mathop {\lim }\limits_{\Delta t \to 0} \left( {\frac{{\Delta \theta }}
{{\Delta t}}} \right) = \frac{{d\theta }}
{{dt}}
\)
13.The angular acceleration \(\alpha\) as the time rate of change of angular velocity.
Thus \(
\alpha = \frac{{d\omega }}
{{dt}}
\)
14.Angular velocity of seconds hand
\(
\omega = \frac{{2\pi }}
{t} = \frac{{2\pi }}
{{60}} = \frac{\pi }
{{30}}radS^{ - 1} \)
*Angular velocity of minutes hand
\(
\omega = \frac{{2\pi }}
{{60 \times 60}} = \frac{\pi }
{{1800}}radS^{ - 1}
\)
*Angular velocity of hours hand
\(
\omega = \frac{{2\pi }}
{{12 \times 3600}} = \frac{\pi }
{{21600}}radS^{ - 1}
\)
15..The instantaneous angular acceleration is defined as
\(
\alpha = \mathop {\lim }\limits_{\Delta t \to 0} \frac{{\Delta \omega }}
{{\Delta t}} = \frac{{d\omega }}
{{dt}} = \frac{d}
{{dt}}\left( {\frac{{d\theta }}
{{dt}}} \right)
\) = \(
\frac{{d^2 \theta }}
{{dt^2 }}\)
16.When a particle is moving along a circle of radius r with a uniform speed v, then the centripetal acceleration is ar.
\(
\overrightarrow {a_r } = \overrightarrow \omega \times \overrightarrow v = \overrightarrow \omega \times \left( {\overrightarrow v \times \overrightarrow r } \right)\)
\(
a_r = \omega v = r\omega ^2 = \frac{{v^2 }}
{r} - 4\pi ^2 f^2 r\)
17.Net linear acceleration of particle in circular motion
\(
a = \sqrt {a_c^2 + a_t^2 } \)
\(
a_t = \frac{{dv}}
{{dt}}\) is called tangential acceleration