Engr Help

Kinetics – Constrained Motion of a Particle in a Plane

In a Nut Shell: Euler's First Law( F = m a) governs the motion of a particle in plane motion.

For a rectangular coordinate description the equations of motion are:

Σ F_x = ma_x and Σ F_y = ma_y

In words, the net force in the x-direction acting on the particle equals its mass time its x-component

of acceleration. The same description holds for the y-component. Note that the gravitational force,

mg, typically acts in the ˗ y-direction.

Strategy: Solution for constrained motion of a particle in a plane involves five key steps.

Step 1: Draw a free body diagram showing all external forces acting on the particle. In

general there will be the gravitational force on the particle, the x-component of the external

force acting on the particle, and the y-component of the external force acting on the particle.

Step 2: Write down Euler’s 1^st law for the particle. i.e. F = m a

F_x = m(d²x/dt²) and F_y ˗ mg = m(d²y/dt²)

Step 3: Account for any constraints acting on the particle. The constraints come in two distinct

types - geometric constraints and kinematic constraints.

Type 1: Geometric constraints such as the motion of the particle is constrained to move

along a path or wire such as y(x) = A sin(bx) or y(x) = A cos(bx) .

Type 2: Kinematic constraints come in various forms.

a. dx/dt = constant (positive or negative) dx/dt is the x-component of the velocity

b. v = speed = constant (positive or negative); speed is the magnitude of the velocity

c. dv/dt = change of speed = constant (positive, negative, or zero)

d. and various combination of a, b, and c.