Question

A particle of mass \(m\) is subject to a force \(\boldsymbol{F}\). Obtain the equations of motion in cylindrical polar coordinates.

Short answer

Answer: The equations of motion in cylindrical polar coordinates for a particle of mass m subject to a force F are: 1. ρ-component: m[(d^2ρ/dt^2) - ρ(dφ/dt)^2] = F_ρ 2. φ-component: m[2(dρ/dt)(dφ/dt) + ρ(d^2φ/dt^2)] = F_φ 3. z-component: m(d^2z/dt^2) = F_z

Step-by-step solution

Write the position vector in cylindrical polar coordinates

The position vector in cylindrical polar coordinates can be written as: r(ρ, φ, z) = ρ(cos(φ)î + sin(φ)ĵ) + zk^

Compute the velocity vector in cylindrical polar coordinates

To compute the velocity vector in cylindrical polar coordinates, differentiate the position vector with respect to time: v(ρ, φ, z) = dr/dt = (dρ/dt)(cos(φ)î + sin(φ)ĵ) + ρ(-sin(φ)(dφ/dt)î + cos(φ)(dφ/dt)ĵ) + (dz/dt)k^

Compute the acceleration vector in cylindrical polar coordinates

Now we differentiate the velocity vector with respect to time to obtain the acceleration vector in cylindrical polar coordinates: a(ρ, φ, z) = dv/dt = [(d^2ρ/dt^2) - ρ(dφ/dt)^2] (cos(φ)î + sin(φ)ĵ) + [2(dρ/dt)(dφ/dt) + ρ(d^2φ/dt^2)](-sin(φ)î + cos(φ)ĵ) + (d^2z/dt^2)k^

Express the force in cylindrical polar coordinates

Now we need to express the force F in terms of its components in the cylindrical polar coordinates. The force can be written as: F(ρ, φ, z) = F_ρ(cos(φ)î + sin(φ)ĵ) + F_φ(-sin(φ)î + cos(φ)ĵ) + F_zk^

Write the equations of motion in cylindrical polar coordinates

Now we can write the equations of motion by setting the force F equal to the mass m times the acceleration a in cylindrical polar coordinates. This will give us three separate equations, one for each coordinate: Equation 1 (ρ-component): m[(d^2ρ/dt^2) - ρ(dφ/dt)^2] = F_ρ Equation 2 (φ-component): m[2(dρ/dt)(dφ/dt) + ρ(d^2φ/dt^2)] = F_φ Equation 3 (z-component): m(d^2z/dt^2) = F_z These are the equations of motion for the particle of mass m subject to the force F in cylindrical polar coordinates.