A particle (p) moves along the curved surface (s). Show that p will remain in contact with s provided that at all times its speed v≥√rgcosteta.
Draw a figure. look at the forces :
centripetal force =mv^2/radius
gravity force along the radius: mg*CosTheta (angles measured from the top).
set them such that gravit>=cent force
v^2/r>=g*cosTheta
solve for v
To show that the particle will remain in contact with the curved surface, we need to demonstrate that the speed of the particle is always greater than or equal to the magnitude of the component of the gravitational force directed towards the surface.
Let's break down the problem:
1. Parameterize the curved surface: Let's assume that the curved surface can be parameterized by two variables, θ and ϕ. The position vector of the particle on the surface can be expressed as r(θ, ϕ).
2. Determine the gravitational force: Since the particle is subject to the force of gravity, we need to find the gravitational force acting on the particle. The gravitational force is given by F = -m * g * n, where m is the mass of the particle, g is the acceleration due to gravity, and n is the unit vector that points towards the gravitational center.
3. Determine the direction of the gravitational force: To determine the component of the gravitational force directed towards the surface, we need to find the unit normal vector at the point on the surface closest to the particle. This can be done by taking the gradient of the position vector r(θ, ϕ) and normalizing it.
4. Calculate the component of the gravitational force: Once we have the unit normal vector, we can find the magnitude of the component of the gravitational force directed towards the surface. This can be obtained by taking the dot product of the gravitational force vector and the unit normal vector.
5. Express the particle's speed: Finally, we need to express the particle's speed in terms of the given variables. The speed of the particle can be calculated by taking the magnitude of the derivative of the position vector with respect to time, |v| = |dr/dt|.
6. Compare the speed and gravitational force: Comparing the magnitude of the particle's speed to the magnitude of the component of the gravitational force directed towards the surface, we need to show that |v| ≥ √(r * g * cos(θ)).
By following these steps, we can demonstrate whether the particle will remain in contact with the curved surface based on the given condition.