The image provides several fundamental equations from physics, covering various concepts such as force, motion, energy, and more. Here’s a breakdown of each equation:
1. Force:
- \(\Sigma F = \frac{dp}{dt} = \frac{d(mv)}{dt}\): This is the equation for force based on the rate of change of momentum (\(p = mv\)).
- \(\Sigma F = ma\): Newton’s Second Law of Motion for constant mass, where \(F\) is the force, \(m\) is the mass, and \(a\) is the acceleration.
2. Acceleration:
- \(\text{a}_{\text{average}} = \frac{\Delta v}{\Delta t}\): Average acceleration is the change in velocity (\(\Delta v\)) over the change in time (\(\Delta t\)).
- \(a = \frac{dv}{dt} = \frac{d^2 s}{dt^2}\): Instantaneous acceleration is the derivative of velocity or the second derivative of displacement with respect to time.
3. Impulse:
- \(J = \Delta p = \int F dt\): Impulse is the change in momentum, which is also the integral of force over time.
- \(J = F \Delta t\): If the force is constant, impulse can be calculated as the product of force and time.
4. Velocity:
- \(V_{\text{average}} = \frac{\Delta d}{\Delta t}\): Average velocity is the change in distance (\(\Delta d\)) over the change in time.
- \(v = \frac{ds}{dt}\): Instantaneous velocity is the derivative of displacement with respect to time.
5. Kinetic Energy:
- \(T = \frac{1}{2} mv^2\): Kinetic energy is the energy associated with an object in motion, where \(m\) is the mass and \(v\) is its velocity.
6. Gravity:
- \(F = \frac{Gm_1 m_2}{r^2}\): This is Newton’s Law of Universal Gravitation, where \(G\) is the gravitational constant, \(m_1\) and \(m_2\) are masses, and \(r\) is the distance between them.
7. Mass-Energy (Einstein's equation):
- \(E = mc^2\): This equation describes the equivalence of mass and energy, where \(E\) is energy, \(m\) is mass, and \(c\) is the speed of light in a vacuum.
8. Density:
- \(\rho = \frac{m}{v}\): Density is mass divided by volume.
9. Motion (Kinematic Equations):
- \(v = v_0 + at\): Final velocity is initial velocity plus acceleration times time.
- \(s = v_0 t + \frac{1}{2} at^2\): Displacement is initial velocity times time plus half the product of acceleration and the square of time.
- \(v^2 = v_0^2 + 2as\): Final velocity squared is initial velocity squared plus twice the product of acceleration and displacement.
10. Torque:
- \(\Sigma \tau = \frac{dL}{dt}\): Torque is the rate of change of angular momentum (\(L\)).
- \(\Sigma \tau = r \times F\): Torque is the cross product of position vector \(r\) and force \(F\).
11. Variance:
- \(s^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \bar{x})^2\): Variance measures the dispersion of a set of data points from their mean, \(x_i\) represents each data point, and \(\bar{x}\) is the mean.
12. Charge:
- \(Q = It\): The charge \(Q\) is the product of current \(I\) and time \(t\).
13. Drude Law:
- \(\alpha = \frac{k}{\lambda^2 - \lambda_0^2}\): This relates to the interaction of light with matter, where \(k\) is a constant, \(\lambda\) is the wavelength, and \(\lambda_0\) is a reference wavelength.
These are some of the key equations used to solve problems in classical mechanics, electromagnetism, and thermodynamics.