Circular motion - marble in bowl

[Thanuka]

Can anyone please answer this question

1.A glass marble of mass m is moving int a horizontal circle round the inside surface of a smooth hemispherical bowl of radius r. The centre of the circle is at a distance 1/2r below the centre of the bowl. (i) Find the magnitude of the reaction between the marbel and the bowl (ii) and the speed of the marble.

 

[rl.bhat]

Note down the two forces acting on the marble.
Marble is moving in a horizontal circle without slipping along the bowl.
So identify equal and opposite camponents of the two forces acting on the marble.
Other camponents of the two forces adds up to give the normal reaction.

 

[baba89]

I would like to address this question by first acknowledging the scenario described - a marble moving in a horizontal circle inside a smooth hemispherical bowl. This type of motion is known as circular motion, where an object moves along a circular path at a constant speed. In this case, the marble is moving with a constant speed around the inside surface of the bowl, with the centre of the circle located at a distance of 1/2r below the centre of the bowl.

Now, to answer the first part of the question, we need to consider the forces acting on the marble. Since the marble is moving in a circular path, there must be a centripetal force acting towards the centre of the circle, keeping the marble in its circular motion. This force is provided by the reaction force between the marble and the bowl. The magnitude of this reaction force can be calculated using the following equation: F = mv^2/r, where m is the mass of the marble, v is its speed, and r is the radius of the circular path. Therefore, the magnitude of the reaction force would be F = m(v^2/r).

Moving on to the second part of the question, we can use the same equation to find the speed of the marble. Rearranging the equation, we get v = √(Fr/m). Substituting the values of F and r from the given information, we get v = √(m/2), where m is the mass of the marble. Therefore, the speed of the marble would be directly proportional to the square root of its mass.

In conclusion, the magnitude of the reaction force between the marble and the bowl would be m(v^2/r), and the speed of the marble would be √(m/2). I hope this response helps in understanding the concept of circular motion and its application in this scenario.