#### Dynamics - Momentum and Circular Motion

__Newton’s Laws of Motion__

__Newton’s Laws of Motion__

**First Law**: If a body is at rest, it remains at rest or of it is in motion, it moves with a uniform velocity until it is acted on by a resultant force or torque.**Second Law**: The rate of change of momentum of a body is proportional to the resultant force and occurs in the direction of the force;*F*=*ma***Third Law**: If a body A exerts a force on a body B, then body B exerts an equal but opposite force on body A, forming an action-reaction pair.

__Mass and Weight__

__Mass and Weight__

**Mass**is a measure of the amount of matter in a body, and is the property of a body which resists a change in motion.**Weight**is the force of gravitational attraction (exerted by the Earth) on a body.

**MOMENTUM**

- Linear momentum is the product of mass and velocity;
*p*=*mv* - Force is the rate of change of momentum;
*F*=^{(mv – mu)}/_{t} - Impulse is the product of force and the time for which it acts;
*Ft*=*mv*–*mu* - Principle of Conservation of Linear Momentum occurs when bodies in a system interact, the total momentum remains constant provided there is no external force acting on the system;

*m*+_{A}u_{A}*m*=_{B}u_{B}*m*+_{A}v_{A}*m*_{B}v_{B}

__Elastic Collisions__

__Elastic Collisions__

- Momentum is conserved.
- Kinetic energy is conserved
- For example:
- Two identical spheres collide elastically. Initially, X is moving with speed
*v*and Y is stationary. What happens after the collision?

- relative velocity before collision = -(relative velocity after collision)
*u*–_{A}*u*=_{B}*v*–_{B}*v*_{A}

- Two identical spheres collide elastically. Initially, X is moving with speed

__Inelastic Collisions__

__Inelastic Collisions__

- Relative speed of approach > Relative speed of separation
- Momentum is conserved.
- Perfectly Inelastic Collision: In this form of collision, only momentum is conserved, and the particles stick together after the collision, i.e. move with the same velocity.
- In inelastic collision, energy is conserved but E
_{k}may be converted into other forms, e.g. heat.

__Collisions in Two Dimensions__

__Collisions in Two Dimensions__

- The change in momentum (impulse) affecting each sphere acts along the line of impact.
- The law of conservation of momentum applies along the line of impact.
- The components of velocities of the spheres along the plane of impact remain unchanged.

**CIRCULAR MOTION**

- A body moving in a circle at a constant speed changes its velocity since its direction changes. Thus, it is accelerating and hence experiences a force.
**Centripetal Force**is the resultant force acting on an object moving in a circle, it is always directed towards the center of the circle.*F*=^{mv2}/_{r}=*mrω*^{2}**Centripetal Acceleration**is derived by equating Newton’s second law and centripetal force;*a*=*rω*or^{2}*a*=^{v2}/_{r}

__Examples__

__Examples__

- A horizontal flat plate is free to rotate about a vertical axis through its center. A mass M is placed on the plate, a distance d, 35 cm, from the axis of rotation. The speed of rotation is increased from zero until the mass slides off the plate. The maximum frictional force F between the plate and the mass is given by the expression:
*F*=*0.72 W*Determine the maximum number of revolutions per minute for the mass M to remain on the plate

- The centripetal force on the particle is the frictional force so the maximum speed is when friction is at maximum:
- Centripetal Force = Frictional Force
- M
^{v2}/_{r}= 0.72 W - Manipulating the expression by adding ω and removing M: M
^{(ωr)2}/_{r}= 0.72 Mg - ω
^{2}r = 0.72 g

- Find the angular velocity: ω = √(
^{[0.72 × 9.81]}/_{0.35}) = 4.49 s^{-1} - Find the radians covered in a minute using ratios:
- 4.49 rad = 1 sec
- 269.5 rad = 60 secs

- Divide the radians covered by 2π to find revolutions:
^{269.5}/_{2π}= 42.9- ∴ 42 revolutions min
^{-1}

- The centripetal force on the particle is the frictional force so the maximum speed is when friction is at maximum: