Applying Newton's Second Law

 

Many problems will require the use of both kinetics and kinematics to solve.

 

1. Solve the kinetics problem (∑F = ma) to find the acceleration and then solve the kinematics problem (a = dv/dt, v = dr/dt) to find the velocity, displacement or time.

 

2. Solve the kinematics problem (a = dv/dt, v = dr/dt) to find the acceleration and then solve the kinetics problem (∑F = ma) to find the force.

 

Solving Newtonian mechanics problems does not limit you to a particular coordinate system. However, the choice of coordinate frame can greatly affect the ease with which a solution is obtained.

 

Newton's second law in rectangular coordinates

 

Useful when ...

• x- and y-direction accelerations are independent. (e.g. projectile motion)
• motion only occurs in one direction.

 

∑F_x = ma_x

 

∑F_y = ma_y

 

Newton's second law in normal-tangential coordinates

 

The n-t coordinate system has a third axis. This third axis is the bi-normal axis (b – axis). The b – axis is perpendicular to both the n and t axes.

 

Useful when ...

• the particle’s path (radius of curvature) is known.
• information about the path of motion is given.

 

∑F_t = ma_t = m(dv/dt)        Acceleration (a_t) is due to the particle either speeding up or slowing down.

 

∑F_n = ma_n = mv²/ρ        Acceleration (a_n) is due to the particle changing direction of motion, where F_n = Centripetal Force.

 

∑F_b = 0        By definition of this coordinate system, a_b = 0 since the particle accelerates only in the n-t plane.

 

Newton's second law in polar coordinates

 

Useful when ...

• you are given or desire information about the body’s motion relative to a fixed reference frame.
• the particle moves in a curved path.

 

∑F_r = ma_r = m((d ²r/dt ²) - rθ)

 

∑F_θ = ma_θ = m(r(d ²θ/dt ²) + 2(dr/dt)(dθ/dt))