4.13 Summary

SUMMARY

• Scalar quantities are quantities with magnitudes only. Examples are distance, speed, mass, and temperature.
• Vector quantities are quantities with magnitude and direction both. Examples are displacement, velocity, and acceleration. They obey the special rules of vector algebra.
• A vector A multiplied by a real number \(\lambda\) is also a vector, whose magnitude is \(\lambda\) times the magnitude of the vector \(\mathbf{A}\) and whose direction is the same or opposite depending upon whether \(\lambda\) is positive or negative.
• Two vectors \(\mathbf{A}\) and \(\mathbf{B}\) may be added graphically using the head-to-tail method or parallelogram method.
• Vector addition is commutative :
  \(\mathbf{A}+\mathbf{B}=\mathbf{B}+\mathbf{A}\)
  It also obeys the associative law:
  \((\mathbf{A}+\mathbf{B})+\mathbf{C}=\mathbf{A}+\mathbf{( B}+\mathbf{C})\)
• A null or zero vector is a vector with zero magnitude. Since the magnitude is zero, we don’t have to specify its direction. It has the properties :
  \(
  \begin{aligned}
  \mathbf{A}+\mathbf{0} &=\mathbf{A} \\
  \lambda \mathbf{0} &=\mathbf{0} \\
  0 \mathbf{A} &=\mathbf{0}
  \end{aligned}
  \)
• The subtraction of vector \(\mathbf{B}\) from \(\mathbf{A}\) is defined as the sum of \(\mathbf{A}\) and \(-\mathbf{B}\) :
  \(
  \mathbf{A}-\mathbf{B}=\mathbf{A}+(-\mathbf{B})
  \)
• A vector \(\mathbf{A}\) can be resolved into component along two given vectors \(\mathbf{a}\) and \(\mathbf{b}\) lying in the same plane:
  \(\mathbf{A}=\lambda \mathbf{a}+\mu \mathbf{b}\)
  where \(\lambda\) and \(\mu\) are real numbers.
• A unit vector associated with a vector \(\mathbf{A}\) has magnitude 1 and is along the vector \(\mathbf{A}\) :
  \(\hat{\mathbf{n}}=\frac{\mathbf{A}}{|\mathbf{A}|}\)
  The unit vectors \(\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}}\) are vectors of unit magnitude and point in the direction of the \(x-y\)-, and \(z\)-axes, respectively in a right-handed coordinate system.
•  A vector \(\mathbf{A}\) can be expressed as
  \(\mathbf{A}=A_{x} \hat{\mathbf{i}}+A_{y} \hat{\mathbf{j}}\)
  where \(A_{x}, A_{y}\) are its components along \(x\)-, and \(y\)-axes. If vector \(\mathbf{A}\) makes an angle \(\theta\) with the \(x\)-axis, then \(A_{x}=A \cos \theta, A_{y}=A \sin \theta\) and \(A=|\mathbf{A}|=\sqrt{A_{x}^{2}+A_{y}^{2}}, \tan \theta=\frac{A_{y}}{A_{x}}\).
• Vectors can be conveniently added using analytical method. If sum of two vectors \(\mathbf{A}\) and \(\mathbf{B}\), that lie in \(x-y\) plane, is \(\mathbf{R}\), then:
  \(
  \mathbf{R}=R_{x} \hat{\mathbf{i}}+R_{y} \hat{\mathbf{j}} \text {, where, } R_{x}=A_{x}+B_{x} \text {, and } R_{y}=A_{y}+B_{y}
  \)
• The position vector of an object in \(x-y\) plane is given by \(\mathbf{r}=x \hat{\mathbf{i}}+y \hat{\mathbf{j}}\) and the displacement from position \(\mathbf{r}\) to position \(\mathbf{r}\) ‘ is given by
  \(
  \begin{aligned}
  &\Delta \mathbf{r}=\mathbf{r}^{\prime}-\mathbf{r} \\
  &=\left(x^{\prime}-x\right) \hat{\mathbf{i}}+\left(y^{\prime}-y\right) \hat{\mathbf{j}} \\
  &=\Delta x \hat{\mathbf{i}}+\Delta y \hat{\mathbf{j}}
  \end{aligned}
  \)
• If an object undergoes a displacement \(\Delta \mathbf{r}\) in time \(\Delta t\), its average velocity is given by \(\mathbf{v}=\frac{\Delta \mathbf{r}}{\Delta t}\). The velocity of an object at time \(t\) is the limiting value of the average velocity
  as \(\Delta t\) tends to zero:
  \(\mathbf{v}=\lim _{\Delta \mathrm{t} \rightarrow 0} \frac{\Delta \mathbf{r}}{\Delta t}=\frac{\mathrm{d} \mathbf{r}}{\mathrm{d} t}\). It can be written in unit vector notation as : \(\mathbf{v}=v_{x} \hat{\mathbf{i}}+v_{y} \hat{\mathbf{j}}+v_{z} \hat{\mathbf{k}} \quad\) where \(\quad v_{x}=\frac{\mathrm{d} x}{\mathrm{~d} t}, v_{y}=\frac{\mathrm{d} y}{\mathrm{~d} t}, v_{z}=\frac{\mathrm{d} z}{\mathrm{~d} t}\)
  When the position of an object is plotted on a coordinate system, \(\mathbf{v}\) is always tangent to the curve representing the path of the object.
• If the velocity of an object changes from \(\mathbf{v}\) to \(\mathbf{v}^{\prime}\) in time \(\Delta t\), then its average acceleration is given by: \(\overline{\mathbf{a}}=\frac{\mathbf{v}-\mathbf{v}^{\prime}}{\Delta t}=\frac{\Delta \mathbf{v}}{\Delta t}\)
  The acceleration \(\mathbf{a}\) at any time \(t\) is the limiting value of \(\overline{\mathbf{a}}\) as \(\Delta t \rightarrow 0\):
  \(
  \mathbf{a}=\lim _{\Delta t \rightarrow 0} \frac{\Delta \mathbf{v}}{\Delta t}=\frac{\mathrm{d} \mathbf{v}}{\mathrm{d} t}
  \)
  In component form, we have : \(\mathbf{a}=a_{x} \hat{\mathbf{i}}+a_{y} \hat{\mathbf{j}}+a_{z} \hat{\mathbf{k}}\) where, \(a_{x}=\frac{d v_{x}}{d t}, a_{y}=\frac{d v_{y}}{d t}, a_{z}=\frac{d v_{z}}{d t}\)
• If an object is moving in a plane with constant acceleration \(a=|\mathbf{a}|=\sqrt{a_{x}^{2}+a_{y}^{2}}\) and its position vector at time \(t=0\) is \(\mathbf{r}_{0}\), then at any other time \(t\), it will be at a point given by:
  \(
  \mathbf{r}=\mathbf{r}_{\mathbf{o}}+\mathbf{v}_{\mathrm{o}} t+\frac{1}{2} \mathbf{a} t^{2}
  \)
  and its velocity is given by:
  \(
  \mathbf{v}=\mathbf{v}_{\mathbf{o}}+\mathbf{a} t
  \)
  where \(\mathbf{v}_{\mathrm{o}}\) is the velocity at time \(t=0\)
  In component form:
  \(
  x=x_{o}+v_{a x} t+\frac{1}{2} a_{x} t^{2}
  \)
  \(
  y=y_{0}+v_{o y} t+\frac{1}{2} a_{y} t^{2}
  \)
  \(
  \begin{aligned}
  &v_{x}=v_{o x}+a_{x} t \\
  &v_{y}=v_{o y}+a_{y} t
  \end{aligned}
  \)
  Motion in a plane can be treated as the superposition of two separate simultaneous one-dimensional motions along two perpendicular directions
• An object that is in flight after being projected is called a projectile. If an object is projected with initial velocity \(\mathbf{v}_{\mathrm{o}}\) making an angle \(\theta_{0}\) with \(x\)-axis and if we assume its initial position to coincide with the origin of the coordinate system, then the position and velocity of the projectile at time \(t\) are given by:
  \(
  \begin{aligned}
  &x=\left(v_{o} \cos \theta_{o}\right) t \\
  &y=\left(v_{o} \sin \theta_{o}\right) t-(1 / 2) g t^{2} \\
  &v_{x}=v_{\alpha x}=v_{o} \cos \theta_{o} \\
  &v_{y}=v_{o} \sin \theta_{o}-g t
  \end{aligned}
  \)
  The path of a projectile is parabolic and is given by:
  \(
  y=\left(\tan \theta_{0}\right) x-\frac{g x^{2}}{2\left(v_{o} \cos \theta_{o}\right)^{2}}
  \)
  The maximum height that a projectile attains is:
  \(
  h_{m}=\frac{\left(v_{o} \sin \theta_{o}\right)^{2}}{2 g}
  \)
  The time taken to reach this height is:
  \(
  t_{m}=\frac{v_{o} \sin \theta_{o}}{g}
  \)
  The horizontal distance travelled by a projectile from its initial position to the position it passes \(y=0\) during its fall is called the range, \(R\) of the projectile. It is:
  \(
  R=\frac{v_{o}^{2}}{g} \sin 2 \theta_{o}
  \)
• When an object follows a circular path at a constant speed, the motion of the object is called uniform circular motion. The magnitude of its acceleration is \(a_{c}=v^{2} / R\). The direction of \(a_{c}\) is always towards the centre of the circle.

The angular speed \(\omega\), is the rate of change of angular distance. It is related to velocity \(v\) by \(v=\omega R\). The acceleration is \(a_{c}=\omega^{2} R\).

If \(T\) is the time period of revolution of the object in circular motion and \(v\) is its frequency, we have \(\omega=2 \pi v, v=2 \pi v R, a_{c}=4 \pi^{2} v^{2} R\)