## Laws of Motion

# Second Law of Motion: Momentum and Impulse

- According to second law of motion the rate of change of linear momentum is directly proportional to the external force applied on it.
- Newton’s second law of motion gives the formula of force f = ma.
- Second law implies that when a bigger force is applied on a body its linear momentum changes fast.
- Linear momentum is defined as the product of mass and velocity P = mv.
- Momentum in a vector quantity.
- Relation between F and P is \tt F \propto \frac{dp}{dt}
- 1 Newton = 10
^{5}dynes. - \tt F = \frac{m(v - u)}{t} where, v = final velocity, u = inertial velocity.
- If “v” is constant and ‘m’ is changing \tt F = V\frac{dm}{dt}
- \tt F = \frac{P_{2} - P_{1}}{t} where, P
_{1}= initial momentum, P_{2}= final momentum - When a large force is acting on a small internal of time the product of force and time is called Impulse (J).
- The impulse J = f(t) = ∫fdt is a quantity that combines the net force and the time interval over which the force acts.
- Area under force time graph gives impulse.
- If a gun fires n bullets of mass ‘m’ \tt F = \frac{nmv}{t}
- If water of density ‘ρ’ coming out of pipe of area of cross section A with speed ‘v’ F = Aρv
^{2} - The reaction force on a person in a lift moving up with acceleration R = m (g + a)

- The reaction force on a person in a lift moving down with deceleration R = m (g + a)
- The reaction force on a person in a lift moving up deceleration R = m (g − a)
- The reaction force on a person in a lift moving down acceleration R = m (g − a)
- The reaction force on a person in a freely falling lift R = 0.
- The force on between two bodies at contact is called contact force.
- Contact force \tt F = \frac{M_{2} F}{M_{1} + M_{2}}

- Acceleration of two bodies system \tt a = \frac{F}{M_{1} + M_{2}}
- For three bodies \tt a = \frac{F}{m_{1} + m_{2} + m_{3}}

- Tension is an electromagnetic force in a string due to force.
- Acceleration of two bodies connected by string \tt a = \frac{F}{m_{1} + m_{2}}
- Tension in string \tt T = \frac{m_{2} F}{m_{1} + m_{2}}
- Acceleration of three bodies connected by string \tt a = \frac{F}{m_{1} + m_{2} + m_{3}}
- Tension in first string \tt T_{1} = \frac{m_{1} F}{m_{1} + m_{2} + m_{3}}
- Tension in second string \tt T_{2} = \frac{\left(m_{1} + m_{2}\right) F}{m_{1} + m_{2} + m_{3}}
- At woods machine acceleration \tt a = \frac{m_{1} - m_{2}}{m_{1} + m_{2}} g

- Tension \tt T = \frac{2 m_{1}m_{2}}{m_{1} + m_{2}} \cdot g
- Thrust on the pulley = \tt 2T = \frac{4 m_{1}m_{2} g}{m_{1} + m_{2}}
- Acceleration \tt a = \frac{m_{1} g}{m_{1} + m_{2}}

- Tension in string \tt T = \frac{m_{1} m_{2} g}{m_{1} + m_{2}}
- Acceleration \tt a = \frac{\left(m_{1} - m_{2} \sin \theta \right) g}{m_{1} + m_{2}}

- Tension \tt T = \frac{m_{1} m_{2} \left(1 + \sin \theta \right)}{m_{1} + m_{2}}
- Acceleration \tt a = \frac{\left(m_{1} \sin \alpha - m_{2} \sin \beta \right) g}{m_{1} + m_{2}}

- Tension \tt T = \frac{m_{1} m_{2} \left(\sin \alpha + \sin \beta \right) g}{m_{1} + m_{2}}

### View the Topic in this video From 0:36 To 44:05

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1. Newton's Second Law of Motion: \tt F = \frac{mdv}{dt} = ma

2. Impulse = Force × Time = Change in momentum

3. Position dependent force: Gravitational force between two bodies \frac{Gm_{1}m_{2}}{r^{2}}