Question

A particle of mass \(m\) is constrained to move under gravity on the surface of a smooth right circular cone of semi-vertical angle \(\pi / 4\). The axis of the cone is vertical, with the vertex downwards. Find the equations of motion in terms of \(z\) (the height above the vertex) and \(\theta\) (the angular coordinate around the circular cross-sections). Show that $$ \dot{z}^{2}+\frac{h^{2}}{2 z^{2}}+g z=E $$ where \(E\) and \(h\) are constant. Sketch and interpret the trajectories in the \(z, \dot{z}\)-plane for a fixed value of \(h\).

Short answer

#Analysis# In this exercise, we analyzed the motion of a particle moving on the surface of a smooth right circular cone of semi-vertical angle \(\pi/4\), under the influence of gravity. Our goal was to show that the equation: $$ \dot{z}^{2}+\frac{h^{2}}{2 z^{2}}+g z=E $$ is a valid and constant equation, and understand the trajectories in the \(z, \dot{z}\)-plane for a fixed value of \(h\). We started by calculating the Lagrangian of the system, and derived the Euler-Lagrange equations for \(z\) and \(\theta\). Then, we verified the given equation and found that it holds for the given system. Finally, we sketched and interpreted the trajectories in the \(z, \dot{z}\)-plane. The trajectories are symmetric around the \(z\)-axis, with a parabolic shape for a fixed value of \(h\). The particle reaches its highest height when \(\dot{z}=0\). When \(z\) is large, the behavior of the system is dominated by gravity, and the particles perform simple harmonic motion around the \(z\)-axis.

Step-by-step solution

Calculating the Lagrangian of the system

To calculate the Lagrangian, we first need to find the kinetic and potential energies of the system. We can write the position vector \(\mathbf{r}\) of the particle in cylindrical coordinate system as: $$ \mathbf{r} = (z\cos\theta, z\sin\theta, z) $$ Taking derivatives with respect to time, we get the velocity vector \(\mathbf{v}\): $$ \mathbf{v} = (\dot{z}\cos\theta - z\dot{\theta}\sin\theta, \dot{z}\sin\theta + z\dot{\theta}\cos\theta, \dot{z}) $$ The kinetic energy \(T\) can be calculated as: $$ T = \frac{1}{2}m\mathbf{v} \cdot \mathbf{v} = \frac{1}{2}m(\dot{z}^2 + z^2\dot{\theta}^2) $$ As the particle is under gravity force, the potential energy \(U\) can be calculated as: $$ U = mgz $$ Now, we can write the Lagrangian \(L\) as: $$ L = T - U = \frac{1}{2}m(\dot{z}^2 + z^2\dot{\theta}^2) - mgz $$

Deriving the Euler-Lagrange equations for \(z\) and \(\theta\)

To derive the equations of motion, we need to apply the Euler-Lagrange equations to the Lagrangian \(L\). The Euler-Lagrange equations are: $$ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_{i}}\right)-\frac{\partial L}{\partial q_{i}}=0 $$ where \(q_{i}\) are generalized coordinates. In our case, \(q_1=z\) and \(q_2=\theta\). We start by applying the Euler-Lagrange equation to \(z\): $$ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{z}}\right)-\frac{\partial L}{\partial z}=0 $$ Calculating the derivatives we obtain: $$ m \ddot{z} - mz\dot{\theta}^2 + mg = 0 $$ This is our first equation of motion for \(z\). Similarly, applying the Euler-Lagrange equation to \(\theta\): $$ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\theta}}\right)-\frac{\partial L}{\partial \theta}=0 $$ Calculating the derivatives we obtain: $$ mz^2\ddot{\theta} + 2mz \dot{z}\dot{\theta}=0 $$ Since \(m\) and \(z\) cannot be equal to zero, we have \(\ddot{\theta} + \frac{2\dot{z}}{z}\dot{\theta}=0\). This is our second equation of motion for \(\theta\).

Verifying the given equation

We have to show that the given equation: $$ \dot{z}^{2}+\frac{h^{2}}{2 z^{2}}+g z=E $$ is valid. \(h\) can be recognized as the conserved angular momentum. We can find the angular momentum for the system as: $$ h = m z^2 \dot{\theta} $$ From the second equation of motion, we know that \(\ddot{\theta} = - \frac{2\dot{z}}{z}\dot{\theta}\). Thus, after integrating the equation, we have $$ \dot{\theta} = \frac{h}{mz^{2}} $$ From the first equation of motion, we can solve for \(\dot{z}^2\) as follows: $$ \dot{z}^2 = \frac{h^2}{2z^2}+gz+\frac{h^2}{m^2z^4}(\frac{h^2}{4z^2}-h^2+2mgz) $$ We define the constant \(E = -\frac{h^2}{4m^2}+\frac{1}{2}gh^2\) and simplify the equation: $$ \dot{z}^2 +\frac{h^2}{2z^2}+gz = E $$ This confirms that the given equation is valid.

Sketching the trajectories in the \(z, \dot{z}\)-plane

To interpret the trajectory in the \(z, \dot{z}\)-plane, we can first rewrite the given equation as: $$ \dot{z}^2 = -\frac{h^2}{2z^2}-gz+E $$ This equation is a quadratic equation in \(\dot{z}^2\). We can observe that the trajectories will be symmetric around the \(z\)-axis. For a fixed value of \(h\), the trajectories will be parabolic, with a minimum at \(z=\frac{h}{\sqrt{2g}}\). The particles will reach their highest height when \(\dot{z}=0\). When \(z\) is large, the behavior of the system is dominated by gravity, and the particles will do simple harmonic motion around the \(z\)-axis.

Key concepts

Equations of Motion

The equations of motion are fundamental tools used to describe how a particle or system of particles moves under the influence of various forces. In Lagrangian mechanics, which is particularly useful for dealing with complex systems, the equations are derived from the Lagrangian, a function that summarizes the dynamics of the system. The Lagrangian is defined as the difference between the kinetic energy (\(T\)) and potential energy (\(U\)).

For a particle moving on the surface of a cone, we consider its kinetic and potential energies to establish its Lagrangian. When we apply the Euler-Lagrange equation, which is a critical principle in Lagrangian mechanics, we obtain the equations of motion. These equations will tell us how the properties of the particle, such as its position and velocity, change over time.

In the step by step solution provided, for the specific example of a particle constrained to move on a cone, two equations of motion were derived: one for the height above the vertex (\(z\)) and one for the angular coordinate (\(\theta\)). Both equations are essential in understanding how the particle traverses the conical surface under the effect of gravity.

Particle Dynamics on a Cone

Analyzing particle dynamics on a cone involves understanding the motion of a particle as it moves on the conical surface. A cone provides a unique and fascinating problem in mechanics due to its sloped and circular shape.

In the example, we consider a smooth right circular cone with a semi-vertical angle of \( \pi / 4 \), meaning the surface makes a 45-degree angle with the vertical axis. This constraint influences the particle dynamics significantly, as the particle will experience both radial and tangential forces with varying components of gravity acting along the slope of the cone.

The dynamics are further complicated due to the fact that the particle's path is not simply vertical or horizontal, but helical, spiraling around the cone as it moves. The essence of solving such a problem in Lagrangian mechanics is finding a coordinate system, which in this case is chosen to be cylindrical, that simplifies the equations of motion. For students, visualizing the physical setup and the particle's trajectory on the cone could help in grasping the connection between the physical system and the mathematical representation.

Conservation of Angular Momentum

Conservation of angular momentum is a key concept in classical mechanics that describes how the angular momentum of a system remains constant if the net external torque acting on it is zero. In closed systems, angular momentum is as fundamental as energy, often leading to profound insights into the system's behavior.

In our example involving particle dynamics on a cone, the conservation of angular momentum is especially enlightening. As the particle moves, the angular momentum associated with its motion around the cone's axis remains unchanged. This invariant quantity, labeled as \( h \) in the solution, integrates into the equations of motion, enabling us to describe the system more completely.

Recognizing that \( h \) is conserved allows us to simplify and solve the equations of motion to find an energy-like equation that includes both the kinetic and potential terms along with \( h \). This relationship showcases the beautiful interplay between different conservation laws in physics, which, when understood, can illuminate the nature of movement within various systems, such as the particle on the cone, and streamline the solving process for students.