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Chapter 3: Problem 2

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

Short Answer

Expert verified
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

01

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^
02

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^
03

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^
04

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^
05

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.

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Most popular questions from this chapter

A particle \(P\) of mass \(m\) is attached to two light inextensible strings, each of length \(a\). The strings pass over two smooth pegs \(A\) and \(B\), which are at the same height and distance \(2 b\) apart. At the other ends of the strings hang two particles of mass \(m\), which can move up and down the vertical lines through \(A\) and \(B\). The particle \(P\) can move in the vertical plane containing \(A\) and \(B\). Show that if \(2 b \cosh \varphi=P A+P B\) and \(2 b \cos \theta=P A-P B\), then the kinetic energy of \(P\) is $$ T=\frac{1}{2} m b^{2}\left(\cosh ^{2} \varphi-\cos ^{2} \theta\right)\left(\dot{\theta}^{2}+\dot{\varphi}^{2}\right) $$ Hence find the Lagrangian of the system in terms of \(\theta\) and \(\varphi\).

Two particles, each of mass \(m\), are moving under their mutual gravitational attraction, which is given by the potential \(U=-\gamma m / 2 r\), where \(2 r\) is their separation and \(\gamma\) is a constant. Find the equations of motion in terms of the coordinates \(X, Y, Z, r, \theta, \varphi\), where \(X, Y\), and \(Z\) are the Cartesian coordinates of the centre of mass and \(r, \theta\), and \(\varphi\) are the polar coordinates of one particle relative to the centre of mass.

A particle of unit mass is constrained to move on the surface of a unit sphere, but is otherwise free. Show that the dynamical trajectories are great circles traversed at uniform speeds. Show that if \(\gamma\) is a complete circuit of a great circle in time \(t\), then $$ J_{L}(v)=2 \pi^{2} / t $$ Does \(\gamma\) minimize \(J_{L}\) over all curves on the sphere that start and end at a point \(P\) on the equator and take time \(t\) for the round trip from \(P\) back to \(P ?\)

\({ }^{\dagger}\) Two particles \(P_{1}\) and \(P_{2}\) have respective masses \(m_{1}\) and \(m_{2}\) and are attracted to each other by a force with time- independent potential \(U(\boldsymbol{r})\), where \(\boldsymbol{r}\) is the vector from \(P_{1}\) to \(P_{2}\). (1) Show that the motion of the centre of mass is the same as in the case \(U=0\). (2) Show that the motion of \(P_{2}\) relative to \(P_{1}\) is the same as in the case that \(P_{1}\) is fixed and \(P_{2}\) is attracted to \(P_{1}\) by a force with potential $$ V=\frac{m_{1}+m_{2}}{m_{1}} U . $$

A particle of mass \(m\) is free to move in a horizontal plane. It is attached to a fixed point \(O\) by a light elastic string, of natural length \(a\) and modulus \(\lambda\). Show that the tension in the string is conservative and show that the Lagrangian for the motion when \(r>a\) is $$ L=\frac{1}{2} m\left(\dot{r}^{2}+r^{2} \dot{\theta}^{2}\right)-\frac{\lambda}{2 a}(r-a)^{2} $$ where \(r\) and \(\theta\) are plane polar coordinates with origin \(O\).
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