General Physics
- Fundamental, or base, physical quantities:
- Homogeneity of dimensions: For a given equation z = x + y, all the three terms x, y and z must have the same dimension. Thus, [x] = [y]=[z].
- Conversion of units: \(n_1 \mathrm{M}_1^x \mathrm{~L}_1^y \mathrm{~T}_1^z=n_2 \mathrm{M}_2^x \mathrm{~L}_2^y \mathrm{~T}_2^z\), where nl and n2 are the numerical values (or multipliers) of two physical quantities, and M1, M2, L1, L2, T1, and T2 are different units.
- The slope of a curve: For any curve y =f(x), the slope at any point (P) is \(\frac{d y}{d x}\) = tan θ, where θ is the angle which the tangent at P makes with the x-axis.
- Maxima and minima: The slope \(\left(\frac{d y}{d x}\right)\) at a maximum or minimum is zero.
But \(\frac{d^2 y}{d x^2}<0\) for a maximum,
⇒ \(\frac{d^2 y}{d x^2}>0\) for a minimum.
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- The area under a curve expressed by Y
y=f(x)is
⇒ \(A=\int d A=\int y d x\)
⇒ \(\int_{x_1}^{x_2} f(x) d x\)
- Significant digits: The digits in a number that express the precision of its measurement and not just the magnitude of the number are called its significant digits.
- Counting significant digits:
- Some other examples are as followsθ.
- 50 has only one significant digit (5).
- 50.0 has two significant digits (5 and 0).
- 27.120 x1019 has five significant digits (2, 7,1, 2, 0).
- 45.2 +16.730 = 61.930≈ 61.9.
- 346÷22 = 15.727 ≈16 (rounded to two digits).
- Errors in measurement: The difference between the measured value of a physical quantity Q and its true value Q0 is called the error. Thus, the error in Q is ΔQ = Q-Q0.
- Fractional error = \(\frac{\Delta Q}{Q}\)
- Percentage error = \(\frac{\Delta Q}{Q} \times 100 \%\)
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- Propagation of errors:
- \(Q=A^m B^n \Rightarrow \frac{\Delta Q}{Q}=m\left(\frac{\Delta A}{A}\right)+n\left(\frac{\Delta B}{B}\right)\)
- \(Q=\frac{A^m}{B^n} \Rightarrow \frac{\Delta Q}{Q}=m\left(\frac{\Delta A}{A}\right)+n\left(\frac{\Delta B}{B}\right)\)
- \(Q=\frac{A^m B^n}{C^p} \Rightarrow \frac{\Delta Q}{Q}=m\left(\frac{\Delta A}{A}\right)+n\left(\frac{\Delta B}{B}\right)+p\left(\frac{\Delta C}{C}\right)\)
- Vector notation: \(\vec{a}=a_x \hat{i}+a_y \hat{j}+a_z \hat{k}\), where the magnitude of \(\vec{a}\) is
- \(|\vec{a}|=a=\sqrt{a_x^2+a_y^2+a_z^2}\)
- The position vector of the point P(x, y, z) is given by
- \(\overrightarrow{O P}=\vec{r}=x \hat{i}+y \hat{j}+z \hat{k}\)
- The distance between P{xx, yv z1)and Q(x2, y2, z2) is given by
- \(P Q=\left|\overrightarrow{r_2}-\vec{r}_1\right|=\left|\left(x_2-x_1\right) \hat{i}+\left(y_2-y_1\right) \hat{j}+\left(z_2-z_1\right) \hat{k}\right|\)
- \(\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2+\left(z_2-z_1\right)^2}\)
- Unit vector,\(\hat{a}=\frac{\vec{a}}{|\vec{a}|}=\frac{a_x \hat{i}+a_y \hat{j}+a_z \hat{k}}{\sqrt{a_x^2+a_y^2+a_z^2}}\)
- Dot product: \(\hat{a}=\frac{\vec{a}}{|\vec{a}|}=\frac{a_x \hat{i}+a_y \hat{j}+a_z \hat{k}}{\sqrt{a_x^2+a_y^2+a_z^2}}\).
- Cross product: \(\vec{a} \times \vec{b}=(a b \sin \theta) \hat{n}\), where n is the unit vector perpendicular to the film-plane.
In terms of the components of \(\vec{a} \text { and } \vec{b}\),
\(\vec{a} \times \vec{b}=\left|\begin{array}{ccc}
\hat{i} & \hat{j} & \hat{k} \\
a_x & a_y & a_z \\
b_x & b_y & b_z
\end{array}\right|\) - Kinematics—motion in one and two dimensions:
- Average velocity \(\vec{v}_{\mathrm{av}}=\frac{\Delta \vec{r}}{\Delta t}\).
- Instantaneous velocity \(\vec{v}=\frac{d \vec{r}}{d t}\).
- Average acceleration: \(\overrightarrow{a_{\mathrm{av}}}=\frac{\Delta \vec{v}}{\Delta t}\)
- Instantaneous acceleration \(\vec{a}=\frac{d \vec{v}}{d t}\)
- The equations of kinematics with a constant acceleration a are
- \(\vec{v}=\vec{u}+\vec{a} t, \vec{s}=\vec{u} t+\frac{1}{2} \vec{a} t^2 \text { and } v^2=u^2+2 a s\)
- These equations can be expressed in terms of the x-, y- and z-components.
- Projectile motion:
- The position at a time t is given by
\(x=u_x t=(u \cos \theta) t\) and
\(y=u_y t-\frac{1}{2} g t^2\)
\(x \tan \theta-\frac{g x^2}{2 u^2 \cos ^2 \theta}\) - Time of flight T = \(\frac{2 u \sin \theta}{g}\)
- Horizontal range R = \(\frac{u^2 \sin 2 \theta}{g}\)
- Maximum height H = \(\frac{u^2 \sin ^2 \theta}{2 g}\)
- The condition for the range R to be maximum is 0 = 45°.
- The angles of projection (01 and 02) for the same range are complementary, so \(\theta_1+\theta_2=90^{\circ}\)
- The position at a time t is given by
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- Relative velocity: The velocity of A relative to B is \(\vec{v}_{\mathrm{AB}}=\vec{v}_{\mathrm{A}}-\vec{v}_{\mathrm{B}}\)
Its magnitude is \(\left|\vec{v}_{\mathrm{AB}}\right|=\sqrt{v_{\mathrm{A}}^2+v_{\mathrm{B}}^2-2 v_{\mathrm{A}} v_{\mathrm{B}} \cos \theta}\),where θ is the angle between \(\vec{v}_{\mathrm{A}} \text { and } \vec{v}_{\mathrm{B}}\). - Laws of motion:
- Linear momentum \(\vec{p}=m \vec{v}\)
- Force \(\vec{F}_{\text {net }}=\frac{d \vec{p}}{d t}=\frac{d}{d t}(m \vec{v})=m \frac{d \vec{v}}{d t}=m \vec{a}\)
- Forces never exist alone; they exist in pairs—as an action and its reaction. So, \(\vec{F}_{\mathrm{AB}}=-\vec{F}_{\mathrm{BA}}\)
- In the absence of external forces, the net momentum of a system remains conserved.
- Some common forces:
- Gravitational force or weight (W = mg).
- Tension in a string (T): Tension always acts away from the point of contact and has the same magnitude at each point for a light (or massless) string.
- The normal reaction: It is the contact force exerted by a surface on the block in a direction perpendicular to the surface.
- Spring force: A spring always opposes its expansion or compression, and hence exerts a force in the direction opposite to its expansion or compression is given by F = -kx, where k is the spring, or force, constant (SI unit: newton per metre).
- Friction: It is a self-adjusting, tangential contact force which opposes the relative motion between two surfaces in contact. It has the maximum value \(f_{\max }=\mu \delta \mathrm{N}\), where m is the coefficient of friction.
- Pseudo force: It exists in an accelerated frame (noninertial frame) and is given by \(\vec{F}_{\mathrm{ps}}=m\left(-\vec{a}_{\mathrm{fr}}\right)\)
- Constraint relation: It is the relation connecting the coordinates of different objects linked with a system relative to a fixed point in the reference frame.
- Acceleration of an object:
- Sliding down a rough inclined plane, a = \(a=g(\sin \theta-\mu \cos \theta)\).
- Projected up along a rough plane, \(a=\frac{F}{m}-g(\sin \theta+\mu \cos \theta)\)
- Centripetal \(F=m v^2 / r\). Examples of this kind of force are
- The tension in a string whirled with a stone in a circle,
- The gravitational attraction by the sun in planetary motion,
- The electrical attraction in the orbital motion of the electrons in an atom.
- Critical velocity (vc) in a vertical circle:
- At the uppermost point, \(v_c=\sqrt{g r}\).
- At the lowermost point, \(v_c=\sqrt{5 g r}\).
- Banking angle, \(\tan \theta=\frac{v^2}{r g}\).
- Work done by a force:
- Work done by a constant force, \(W=\vec{F} \cdot \vec{s}=F s \cos \theta\).
- Work done by a variable force, \(W=\int \vec{F} \cdot \overrightarrow{d s}\)
- The work done is equal to the area under the F-s graph.
- Kinetic \(E_{\mathrm{k}}=\frac{1}{2} m v^2=\frac{p^2}{2 m}\)
- Work-energy theorem: This theorem states that the total work done on a system due to all forces is equal to the change in the kinetic energy of the system.
- Thus, \(W_{\text {tot }}=\Delta E_{\mathrm{k}}=E_{\mathrm{kf}}-E_{\mathrm{kj}}\)
- Conservative force: A conservative force is one for which the work done by or against it is path-independent and depends only on the starting and finishing points in the motion, and the work done in a closed path is zero.
- Potential energy and potential are defined only for conservative forces. The work done against such forces (external forces) is stored as the potential energy. A system always tends to minimize its potential energy to restore its equilibrium.
- Change in the PE:
- \(\Delta U=U_{\mathrm{f}}-U_{\mathrm{i}}\) = -(work done by the conservative force)
- = \(-\int_i^f \vec{F} \cdot \overrightarrow{d r}\)
- Power = rate of doing work = \(\frac{\Delta W}{d t}\) (SI unit: watt).
- Instantaneous power P = \(P=\frac{d W}{d t}=\frac{d}{d t}(\vec{F} \cdot \overrightarrow{d s})=\vec{F} \cdot \vec{v}\)
- Impulse of a force: \(\vec{J}=\int \vec{F} \cdot d t=\Delta \vec{p}\)= change in the momentum.
- A collision between two objects results in the transfer of momentum and KE between them. The total momentum \((\vec{p})\) always remains constant during all types of collisions.
- Coefficient of restitution (e):
- It is given by the relation: relative velocity of separation = velocity of approach).
- e = 1 for an elastic collision.
- 0 < e <1 for an inelastic collision.
- e = 0 for a perfectly inelastic collision.
- Centre of mass (CM):
- For a system of discrete masses,\(x_{\mathrm{CM}}=\frac{1}{M} \Sigma m_i x_i\)
- For a continuous mass, \(x_{\mathrm{CM}}=\frac{1}{M} \int x d m\)
- The centres of mass of regular bodies are listed below.
- A triangular plate: its centroid.
- A semicircular ring of radius R: \(y_{\mathrm{CM}}=\frac{2 R}{\pi}\)
- A semicircular disc of radius R: \(y_{\mathrm{CM}}=\frac{4 R}{3 \pi}\)
- A hemispherical shell of radius R: \(y_{\mathrm{CM}}=\frac{R}{2}\)
- A solid hemisphere: \(y_{\mathrm{CM}}=\frac{3 R}{8}\)
- A right hollow cone of height H: \(y_{\mathrm{CM}}=\frac{H}{3}\)
- A right solid cone of height H: \(y_{\mathrm{CM}}=\frac{H}{4}\)
- Displacement of the centre of mass:
Since = \(x_{\mathrm{CM}}=\frac{1}{M} \Sigma m_i x_i\) its displacement is
\(\Delta x_{\mathrm{CM}}=\frac{1}{M}\left(m_1 \Delta x_1+m_2 \Delta x_2+\cdots\right)\) \(\frac{1}{M} \Sigma m_i \Delta x_i\) - Velocity of the centre of mass:
\(v_{\mathrm{CM}}=\frac{d x_{\mathrm{CM}}}{d t}=\frac{1}{M}\left(m_1 \frac{d x_1}{d t}+m_2 \frac{d x_2}{d t}+\cdots\right)\)
\(\frac{1}{M} \Sigma m_i v_i\) - Rotational dynamics:
- Angular displacement = θ (in radian).
- Angular velocity ω = \(\omega=\frac{d \theta}{d t}, \text { where } \vec{v}=\vec{\omega} \times \vec{r}\)
- Angular acceleration α = \(\alpha=\frac{d \omega}{d t}, \text { where } \vec{a}=\vec{r} \times \vec{\alpha}\)
- Angular momentum \(\vec{L}\) = \(\vec{L}=\vec{r} \times \vec{p}=I \vec{\omega}\) where I = moment of inertia about the rotational axis.
- Torque \(\vec{\tau}=\vec{r} \times \vec{F}=\frac{d \vec{L}}{d t}=\frac{d}{d t}(I \vec{\omega})=I \vec{\alpha}\)
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- Equations of motion under rotation:
- \(\omega=\omega_0+\alpha t\)
- \(\omega^2=\omega_0^2+2 \alpha \theta\)
- \(\theta=\omega_0 t+\frac{1}{2} \alpha t^2\)
- KE in a rotational motion = \(\frac{1}{2} I \omega^2=\frac{L^2}{2 I}\)
- Conservation of angular momentum:
- When torque = \(\vec{\tau}=\overrightarrow{0}\) angular momentum = \(\vec{L}\) = constant.
- The moment of inertia, \(I=\Sigma m r^2=\int r^2\), is a measure of inertia in rotational motion.
- Condition for pure rolling: \(v_{\mathrm{CM}}=R \omega\)
- KE in rolling motion \(\frac{1}{2} m v_{\mathrm{CM}}^2\left(1+\frac{k^2}{R^2}\right)\),
- where k = radius of gyration about the axis through the CM.
- Acceleration during pure rolling down an inclined plane, \(a=\frac{g \sin \theta}{1+\frac{k^2}{R^2}}\)
- Gravitational force F = \(F=\frac{G m_1 m_2}{r^2}\)
- Relation between and R: \(G M_{\mathrm{B}}=g R_{\mathrm{E}}^2\),
- where ME and RE represent the mass and radius of the earth respectively.
- The magnitude of g:
- At a height h, \(g_h=\frac{g R^2}{(R+h)^2}\)
- At a depth x, \(g_x=g\left(1-\frac{x}{R}\right)\)
- Gravitational field intensity (or strength),
- \(\vec{g}=\frac{\vec{F}}{m}=-\left(\frac{G M_E}{r^2}\right) \hat{r}\).
- The gravitational field strength due to a spherical shell of mass M and radius R at a distance r from the centre is given by
\(|\overrightarrow{\mathscr{G}}|\left\{\begin{array}{l} =\frac{G M}{r^2} \text { for } r>R. \\
=\frac{G M}{R^2} \text { for } r=R . \\
=0 \text { for } r \end{array}\right.\) - Gravitational PE between two masses = \(U=-\frac{G m_1 m_2}{r}\)
- Gravitational potential V = PE per unit mass \(\frac{U}{m}=-\frac{G M}{r}\)
- Relation between potential (V) and field (g): It is given by
\(\mathscr{G}_x=-\frac{\partial V}{\partial x}, \mathscr{G}_y=-\frac{\partial V}{\partial y}, \mathscr{G}_z=-\frac{\partial V}{\partial z}\) - Escape speed = ve = \(\sqrt{2 g R}=\sqrt{\frac{2 G M}{R}}\left(=11.2 \mathrm{~km} \mathrm{~s}^{-1} \text { for earth }\right)\)
- Orbital speed of a satellite = \(v_{\mathrm{o}}=\sqrt{\frac{G M}{r}}\)
- The time of revolution (T) of a satellite may be calculated from
\(T^2=\left(\frac{4 \pi^2}{G M}\right) r^3. \text { Thus, } T^2 \propto r^3\)
- Height of a geostationaryMedical EntrancesatellitePhysicsabove the earth’s surface 4 = 36000 km.
- Simple harmonic motion (SHM):
- Differential equation: \(\frac{d^2 x}{d t^2}+\omega^2 x=0\)
- Displacement x = \(x=A \sin (\omega t+\phi)x=A \sin (\omega t+\phi)\)
- Velocity v = \(\frac{d x}{d t}=\omega A \cos (\omega t+\phi)=\omega \sqrt{A^2-x^2}\)
- Acceleration a = \(\frac{d v}{d t}=-\omega^2 A \sin (\omega t+\phi)=-\omega^2 x\)
- Time period T = \(\frac{2 \pi}{\omega}=2 \pi \sqrt{\frac{\text { displacement }}{\text { acceleration }}}\)
- Kinetic energy = \(\frac{1}{2} m \omega^2 A^2 \cos ^2(\omega t+\phi)=\frac{1}{2} m \omega^2\left(A^2-x^2\right)\)
- Potential energy = \(=\frac{1}{2} m \omega^2 A^2 \sin ^2(\omega t+\phi)=\frac{1}{2} m \omega^2 x^2\)
- Total mechanical energy = \(\frac{1}{2} m \omega^2 A^2\) = constant.
- Time period for a spring-block system \(T=2 \pi \sqrt{\frac{m}{k}}\)
- Time period for a rigid body \(T=2 \pi \sqrt{\frac{I}{m g l}}\)
- Elasticity: It is the property of matter that opposes its deformation and restores its natural shape and size after the deforming force is removed.
- Stress and strain:
- Stress = force per unit area.
- Strain = measure of relative deformation.
- Longitudinal strain = \(\frac{\Delta L}{L}\)
- Volume strains = \(\frac{\Delta V}{V}\)
- Modulus of elasticity = \(\frac{\text { stress }}{\text { strain }}\)
Y (Young modulus), B (bulk modulus) and n (rigidity modulus) are three moduli of elasticity. - Poisson ratio o = \(\frac{\text { lateral strain }}{\text { longitudinal strain }}=\frac{\Delta D / D}{\Delta L / L}\).
where D = diameter and L = length. - The elastic potential energy in a strained body,
\(U=\frac{1}{2} \text { (stress)(strain)(volume) }\)
= \(\frac{1}{2}\) (maximum stretching,force)(extension).
- Surface tension: It is the tendency of a free liquid surface to contract and is measured by the contracting force per unit length.
- Thus, \(S=\frac{F}{l}\).
- Surface energy, U = SxA
- Excess pressure inside a drop, \(\Delta p=\frac{2 S}{R}\)
- Excess pressure inside a soap bubble, \(\Delta p=\frac{4 S}{R}\)
- Excess pressure inside an air bubble, \(\Delta p=\frac{2 S}{R}\)
- Capillary rise = h, where
\(S=\frac{rhpg}{2 \cos \theta}\) - Work done in blowing a bubble W = 8nR2S.
- Viscosity: It is the internal friction in fluid flow.
- Newton’s equation for viscous forces: F = \(\eta A \frac{d v}{d r}\) where,
\(\frac{d v}{d r}\) = velocity gradient, A = area of contact and t = viscosity coefficient. - Stokes’ law: F = \(6 \pi \eta r v\)
- Terminal velocity \(v=\frac{2}{9} \cdot \frac{r^2(\rho-\sigma) g}{\eta}\)
- Critical velocity vc = \(\frac{k \eta}{\rho r}\)
- Equation of continuity = conservation of mass in a fluid flow.
\(A_1 \rho_1 v_1=A_2 \rho_2 v_2\) - For an incompressible fluid (liquid), A1v1 = A2v2.
- Bernoulli equation conservation of energy in fluid flow.
\(p+\rho g h+\frac{1}{2} \rho v^2\) = constant. - Speed of efflux, v = 2gh.
- Gauge pressure = pressure recorded by a gauge. It is given by
\(p_{\text {gauge }}=p_{\text {atm }} \pm h \rho_{\mathrm{m}} g\)
where h = height and pm = density of a liquid in the gauge