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Kinetics of Particles: Work and Energy
Total work done is given by: Modifying this eqn to account for the potential energy terms: U’ 1 - 2 + (- ΔVg ) + (- ΔVe ) = ΔT U’ 1 - 2 is work of all external forces other than the gravitational and spring forces ΔT is the change in kinetic energy of the particle ΔV is the change in total potential energy
- More convenient form because only the end point positions of the particle and end point lengths of elastic spring are of significance. If the only forces acting are gravitational, elastic, and nonworking constraint forces This equation expresses the “Law of Conservation of Dynamical Energy”
U T T T
1 2 2 1
U T V
' 1 2 2 2 '
T 1 V 1 U 1 2 T V
T 1 V 1 T 2 V 2 or E 1 E 2
E = T+V is the total mechanical energy of
the particle and its attached spring
Kinetics of Particles: Work and Energy
Conservation of Energy
- During the motion, only transformation of KE into PE occurs and it can be vice versa A ball of weight W is dropped from a height h above the ground (datum)
- PE of the ball is maximum before it is dropped, at which time its KE is zero. Total mechanical energy of the ball in its initial position is:
- When the ball has fallen a distance h /2, its speed is: Energy of the ball at mid-height position:
- Just before the ball strikes the ground, its PE= and its speed is: The total mechanical energy of the ball: 1 1 2 2 1 2
T V T V or E E
Fixed Origin
Kinetics of Particles :: Impulse and Momentum
Third approach to solution of Kinetics problems
- Integrate the equation of motion with respect to time (rather than disp.)
- Cases where the applied forces act for a very short period of time (e.g., Impact loads) or over specified intervals of time
Linear Impulse and Linear Momentum
Resultant of all forces acting on a particle
equals its time rate of change of linear momentum
Invariability of mass with time!!!
Kinetics of Particles
Linear Impulse and Linear Momentum
Three scalar components of the eqn: Linear Impulse-Momentum Principle
- Describes the effect of resultant force on linear momentum of the particle over a finite period of time Multiplying the eqn by dt ∑ F dt = d G and integrating from t 1 to t 2 The product of force and time is defined as Linear Impulse of the Force. Alternatively: Impulse integral is a vector!! G 1 = linear momentum at t 1 = m v 1 G 2 = linear momentum at t 1 = m v 2 Initial linear momentum of the body plus the linear impulse applied to it equals its final linear momentum
Kinetics of Particles
Linear Impulse and Linear Momentum
Impulsive Forces : Large forces of short duration (e.g., hammer impact)
- In some cases Impulsive forces constant over time they can be brought outside the linear impulse integral. Non-impulsive Forces : can be neglected in comparison with the impulsive forces (e.g., weight of small bodies) In few cases, graphical or numerical integration is required to be performed. The impulse of this force from t 1 to t 2 is the shaded area under the curve Conservation of Linear Momentum If resultant force acting on a particle is zero during an interval of time, the impulse momentum equation requires that its linear momentum G remains constant. The linear momentum of the particle is said to be conserved (in any or all dirn). This principle is also applicable for motion of two interacting particles with equal and opposite interactive forces
Kinetics of Particles: Linear Impulse and Linear Momentum
Example
Solution: Construct the impulse-momentum diagram
Kinetics of Particles: Linear Impulse and Linear Momentum
Example
Solution: The force of impact is internal to the system composed of the block and the bullet. Further, no other external force acts on the system in the plane of the motion. Linear momentum of the system is conserved G 1 = G 2 Final velocity and direction:
Kinetics of Particles
Angular Impulse and Angular Momentum
Velocity of the particle is Momentum of the particle: Moment of the linear momentum vector m v about the origin O is defined as Angular Momentum H O of P about O and is given by: Scalar components of angular momentum: Fixed Origin
Kinetics of Particles
Angular Impulse and Angular Momentum
Rate of Change of Angular Momentum
- To relate moment of forces and angular momentum Moment of resultant of all forces acting on P @ origin: Differentiating with time: v and m v are parallel vectors v x m v = 0 Moment of all forces @ O = time rate of change of angular momentum Scalar components: Fixed Origin Using this vector equation, moment of forces and angular momentum are related
Kinetics of Particles
Angular Impulse and Angular Momentum
Angular Impulse-Momentum Principle
- This eqn gives the instantaneous relation between moment and time rate of change of angular momentum Integrating: The product of moment and time is defined as the angular impulse. The total angular impulse on m about the fixed point O equals the corresponding change in the angular momentum of m about O. Alternatively: Initial angular momentum of the particle plus the angular impulse applied to it equals the final angular momentum.
Kinetics of Particles
Angular Impulse and Angular Momentum
Conservation of Angular Momentum If the resulting moment @ a fixed point O of all forces acting on a particle is zero during an interval of time, the angular momentum of the particle about that point remain constant. Principle of Conservation of Angular Momentum Also valid for motion of two interacting particles with equal and opposite interacting forces
Kinetics of Particles
Angular Impulse and Angular Momentum
Example Solution
Kinetics of Particles
Central Impact
- Can be classified into two types Direct Central Impact (or Direct Impact) - Velocities of the two particles are directed along the line of impact - Direction of motion of the particles will also be along the line of impact Oblique Central Impact (or Oblique Impact) - Velocity and motion of one or both particles is at an angle with the line of impact. - Initial and final velocities are not parallel. Direct Central Impact Oblique Central Impact
Kinetics of Particles: Impact
Direct Central Impact
Collinear motion of two spheres ( v 1 > v 2 )
- Collision occurs with contact forces directed along the line of impact (line of centers)
- Deformation of spheres increases until contact area ceases to increase. Both spheres move with the same velocity.
- Period of restoration during which the contact area decreases to zero
- After the impact, spheres will have different velocities ( v’ 1 & v’ 2 ) with v’ 1 < v’ 2 During impact, contact forces are equal and opposite. Further, there are no impulsive external forces linear momentum of the system remains unchanged. Applying the law of conservation of linear momentum: Assumptions
- Particles are perfectly smooth and frictionless.
- Impulses created by all forces (other than the internal forces of contact) are negligible compared to the impulse created by the internal impact force.
- No appreciable change in position of mass centers during the impact.
