Kinematics of Motion
Introduction
In the previous chapter, we studied that the Theory of Machines deals with the motion of machine parts and the forces acting on them. In this chapter, we focus only on the kinematics of motion, which covers the relative motion of bodies without considering the forces responsible for that motion.
In simple terms, kinematics explains the geometry of motion and important concepts such as displacement, velocity, and acceleration.
Plane Motion
When a body moves in such a way that its motion is restricted to a single plane, it is known as plane motion. Such motion may be rectilinear or curvilinear.
Rectilinear Motion
Rectilinear motion is the simplest form of motion and occurs along a straight line. It is often referred to as translatory motion.
Curvilinear Motion
Curvilinear motion occurs along a curved path. When the motion is restricted to a plane but follows a curved trajectory, it is called plane curvilinear motion.
When particles of a body move in circular paths around a fixed axis, the motion becomes plane rotational motion. A combination of rotation and translation is called general plane motion, which will be discussed in a later chapter.
Linear Displacement
Linear displacement refers to the distance traveled by a body in a specific direction from a fixed reference point. It may occur along a straight line or a curved line.
For example, in a reciprocating engine system, the movement of the piston, connecting rod, and crank generates linear and circular paths whose radius of curvature changes continuously.
Since displacement has both magnitude and direction, it is a vector quantity. Graphically, it can be shown by a straight line.
Linear Velocity
Linear velocity is defined as the rate of change of linear displacement with respect to time. Since it has direction, it is a vector quantity.
Mathematically,
v = ds/dt
Notes:
- If motion is circular, the direction of linear velocity is always tangent to the path.
- Speed is the magnitude of velocity and is scalar.
Linear Acceleration
Linear acceleration is the rate of change of linear velocity with time. It is also a vector quantity.
Mathematically,
a = dv/dt = d²s/dt²
Other forms:
- a = v dv/ds
- If acceleration is negative, it is called retardation.
Equations of Linear Motion
The important equations of motion under uniform acceleration are:
- v = u + at
- s = ut + ½ at²
- v² = u² + 2as
- s = (u + v)/2 × t
Where:
- u = Initial velocity
- v = Final velocity
- a = Constant acceleration
- s = Displacement
- t = Time
Notes:
- If acceleration varies, these equations cannot be directly applied.
- In vertical motion, acceleration becomes g = 9.81 m/s² (downward).
- During upward motion, acceleration becomes –g.
- If a body falls freely from height h, then:
v = √(2gh)