Particle in Sphere Problem: Force Exerted & Speed Calculation

[AK2]

I got a quite interesting problem. ere it goes.
A particle P of mass M,moves on the smooth inner surface of a sphere of a fixed hollow spherical bowl,centre O and inner radius r , describing a horizontal circle at constant speed.the centre C of this circle is at a depth 1/3r vertically below O. Determine
the magintude of force exerted by sphere and speed of p

 

[AK2]

um i made a mistake.i am actually new to this site.

 

[piercas]

The particle in sphere problem is indeed an interesting one. In this scenario, we have a particle moving on the inner surface of a hollow spherical bowl at a constant speed. The particle is held in place by the force exerted by the sphere. To answer the question of the magnitude of this force, we must first understand the dynamics at play.

Since the particle is moving in a horizontal circle, it is experiencing a centripetal force directed towards the center of the circle. This force is provided by the tension in the string that is holding the particle in place. However, in this case, the string is replaced by the force exerted by the sphere.

To determine the magnitude of this force, we can use Newton's second law, which states that the force exerted on an object is equal to its mass multiplied by its acceleration. In this case, the acceleration is the centripetal acceleration, which can be calculated using the formula a = v^2/r, where v is the speed of the particle and r is the radius of the circle.

Now, since the particle is moving at a constant speed, we can also say that the net force acting on it is zero (since there is no change in velocity). Therefore, we can equate the force exerted by the sphere to the centripetal force, and solve for the unknown force.

Once we have determined the magnitude of the force exerted by the sphere, we can use it to calculate the speed of the particle. This can be done using the same formula for centripetal acceleration, but rearranged to solve for the speed v.

In conclusion, the magnitude of the force exerted by the sphere and the speed of the particle can be calculated using the principles of Newtonian mechanics. This problem highlights the importance of understanding the forces at play in a dynamic system and how they can be used to determine the behavior of objects in motion.