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ENGINEERING MECHANICS - DYNAMICS

Engineering mechanics is the science that deals with the state of rest or motion of bodies under the action of forces. It is further divided into mechanics of rigid bodes, deformable bodes and fluids.

Dynamics: deals with bodies in motion. This is further divided to kinetics and kinematics. Kinetics, deals with the bodies in motion due to the application of force, by considering the force that causes the motion. Kinematics, is the study of displacement, velocity and acceleration without considering the force causing the motion. Following are the definition of important terms in dynamics

Speed:

The speed of a body, may be defined as its rate of change of displacement with respect to its surroundings. The speed of a body is irrespective of its direction and hence is a scalar quantity.

Velocity:

The velocity of a body may be defined as its rate of change of displacement, with respect to its surroundings, in a particular direction. As the velocity is always expressed in a particular direction, it is a vector quantity.

Acceleration:

The acceleration of a body may be defined as the rate of change of its velocity. It is said to be positive, when the velocity of a body increase with time, and negative when the velocity decreases with time. The negative acceleration is also called as retardation. In general acceleration denotes the rate at which the velocity is changing. It may be uniform or variable.

Uniform acceleration:

If a body moves in such a way that its velocity changes equal in magnitude in equal intervals of time, it is said to be moving with a uniform acceleration.

Variable acceleration:

If a body moves in such a way that its velocity changes unequal in magnitude in equal intervals of time, it is said to be moving with a variable acceleration.

Types of motion:

Rectilinear motion: When a particle moves in a straight line then it is called as rectilinear motion.

Curvilinear motion: If the particle traces a curve, then curvilinear motion. If the curve lies in a plane, then it is called as plane curvilinear motion.

Uniform motion: A particle in this case should move with a constant velocity and zero acceleration

Uniformly accelerated motion: A particle moving with a constant acceleration is called as uniformly accelerated motion.

Motion with uniform acceleration:

Here 'a' is constant. Hence a = dv / dt

dv = a.dt

( Integrating on both sides, within their limits )

∫ dv = ∫ a dt

( v - u ) = at

v = u + at

v = dx / dt

dx = v dt

Substituting the value of v

dx = ( u + at ) dt

( Integrating on both sides, within their limits )

∫ dx = ∫ ( u + at ) dt = ut + ½ at²

x = ut + ½ at²

a = v dv / dx

a dx = v dv

( Integrating on both sides, within their limits )

a ∫ dx = ∫ v dv

ax = ½ V² - ½ U²

V² = u² + 2ax

Derivation:

Acceleration (a) = Rate of change of velocity with respect to time = dv / dt

Velocity (v) = Rate of change of distance with respect to time = dx / dt

a = dv / dt = d²x / dt²

a = dv / dt = ( dv / dx ) x (dx / dt ) = v dv / dx

Equation of motion:

∑ Fx = ma_x

∑ Fy = ma_y

Fx is the resultant of all forces acting along the x axis.

Fy is the resultant of all forces acting along the y axis.

Equations of dynamic equilibrium:

∑ Fx + ( - ma_x ) = 0

∑ Fy + ( - ma_y ) = 0

The value ( - ma_x ) and ( - ma_y ) is called as inertia force or D' Alembert force.

Curvilinear motion:

The direction of acceleration and velocity may not be the same in curvilinear motion. There are two components of acceleration. They are tangential component ( at ) and normal component ( an )

Tangential component:

at = dv / dt

It is equal to the rate of change of speed of the particle. It is positive and is along the direction of tangent of the motion.

Normal component:

an = v² / ρ

It is the ratio of square of velocity and radius of curvature of the part at that point. The directions is towards the center of curvature of the path. This is also called as the centripetal ( centre seeking ) acceleration.

a = √ ( a_t² + a_n² )

Direction θ = Tan^-1 ( a_n / a_t )

Momentum:

consider the motion of particle of mass 'm' acted by a force F. Then the equation of motion in a generalized from is given as

F = ma = m ( dv / dt ) = d ( mv ) / dt

Thus the force acting on a particle is equal to the rate of change of momentum of the particle. The quantity mv is called as momentum. It unit is Ns.

Impulse:

When a large force acts on a body for a small interval of time then that force is called as impulse force. It can be visualized as the area under the force Vs time diagram. Impulse is nothing but, change in momentum.

Conservation of Momentum:

When the sum of impulses due to external forces is equal to zero, the momentum of the system remains conserved.

Elastic bodies:

The property of a body by virtue of which they rebound after impact is called elasticity. A body which rebounds to a greater height is said to be more elastic and the body that bounces less is called lesser elastic. If a body does not rebound, then it is inelastic. When two bodies collide with each other, the phenomenon of collision takes place as given below.

The body, immediately after collision, come momentarily to rest.
The two bodies tend to compressed each other, so long as they are compressed to the maximum value.
The two bodies attempt to regain its original shape due to their elasticity. This process of regaining the original shape is called as restitution.

The time taken by the two bodies in compression, after the instant of collision, is called as the time of compression and time for which restitution takes place is called the time of restitution. The sum of the two times is called the period of impact or the period of collision.

Impact : The phenomenon of collision of 2 bodies which occurs for a short period of time, during which the two bodies exert a very large force on each other.

Line of Impact: The common normal to the surface of two colliding bodies is called line of impact.

Central / Non-central impact: The centers of body m1 and m2 coincide with Line of impact, hence called as central impact.

Direct / Indirect ( Oblique ) impact: There are the types of collision. If the velocities of two bodies are collinear with line of impact before collision, then is called as direct impact. Else it is indirect impact.

Coefficient of restitution: It is the ratio of the velocity of separation (v2 - v1) and velocity of approach (u1 - u2). Its value lies between 0 to 1. If e = 0 then the two bodies are inelastic. If e = 1, then the two bodies are perfectly elastic.

Projectile:

Any motion which is given just a initial velocity and after which its motion is influenced by acceleration due to gravity is called as projectile. Thus a projectile is moving under the combined effect of vertical and horizontal forces. The vertical component of the motion is always subjected to gravitational acceleration and the horizontal component remains constant. The combined effect of both the forces causes the body to move along a parabolic path. Following are the important terms used in projectiles.

Trajectory is the path traced by a projectile in space.
Velocity of projection is the velocity, with which a projectile is projected.
Angle of projection is the angle with the horizontal, at which a projectile is projected.
Time of flight is the total time taken by the projectile t reach maximum height and to return back to the ground.
Range is the distance, between the point of projection and the point where the projectile strikes the ground.

Equation for the path of projectile is y = x.tana - ( gx² / 2.u².coss²a )

Time of flight of projectile t = 2.u.sin a / g

Horizontal range of projectile R = u².sin 2a / g

Maximum height of projectile H = u². sin² a / 2g

NUMERICAL PROBLEMS

Problem 1: On turning a corner, a motorist rushing at 20 m/s, finds a child on the road 50 m ahead. He instantly stops the engine and applies brake, so as to stop the car within 10 m from the child. Calculate retardation and time required to stop the car.

Solution: Let 'a' be the acceleration. v² = u² + 2as, Here u = 20 m/s and v = 0

0 = (20)² + 2.a.(50 - 10)

a = - 5 m/s² ( Retardation of the car )

We also know that v = u + at. Hence the time required to stop the car is

0 = 20 + (-5) t,

t = 4 seconds.

Problem 2: A stone is dropped from the top of a tower, 50 m high. At the same time another stone is thrown upwards from the foot of the tower with a velocity of 25 m/s. When and where the two stones cross each other?

Solution: Height of the tower is 50 m. First the stone that was dropped from the top is considered. For this u = 0 and a = g. Hence the distance traversed by the stone in time 't' is

s = ut + ½ at² = 0 + 0.5gt²

Now consider the stone that was thrown from the bottom. u = 25 m/s and a = -g. Distance traversed by the stone in time 't' is

50 - s = 25t - 0.5gt²

Adding both the equations we get the value of t = 2 seconds.

The distance at which both stones cross each other is s = 0.5gt² = 0.5 x 9.8 x (2)² = 19.6 m.

Problem 3: A fly wheel runs at a constant 100 rad/s. When the drive motor is switched off the wheel takes 5 minutes to come to rest. What is the angular deceleration?

Solution: Time t = 300 seconds. Initial angular velocity w_o = 100 rad / sec and w = 0.

w = w_o + at,

Hence retardation a = ( 0 - 100 ) / 300 = -0.33 rad / sec²

Problem 4: A racing car takes a bend. Given that vA = 40 m/s , vB = 48 m/s Constant tangential acceleration and R = 300m. What is the total acceleration at B?