A hemispherical bowl of radius R is rotated about its axis of symmetry which is kept vertical. A small block is kept in the bowl at a position where the radius makes an angle Θ with the vertical. The block rotates with the bowl without any slipping. The friction coefficient between the block and the bowl surface is U.Find the range of the angular speed for which the block will not slip.
To find the range of the angular speed for which the block will not slip, we need to consider the forces acting on the block.
1. Gravity: The force due to gravity acts vertically downward. Its magnitude is given by F_gravity = m * g, where m is the mass of the block and g is the acceleration due to gravity.
2. Normal Force: The normal force is the force exerted by the bowl on the block perpendicular to the surface of contact. At any position, the magnitude of the normal force is given by N = m * g * cos(Θ), where Θ is the angle between the radius and the vertical.
3. Friction Force: The friction force acts parallel to the surface of contact between the block and the bowl. Its magnitude is given by F_friction = μ * N, where μ is the coefficient of friction.
For the block to rotate without slipping, the friction force should be less than or equal to the maximum static friction force. The maximum static friction force is given by F_friction_max = μ_s * N, where μ_s is the coefficient of static friction.
Since the block is rotating, there is also a centripetal force acting towards the center of the bowl, given by F_centripetal = m * R * ω^2, where R is the radius of the bowl and ω is the angular speed of rotation.
Now, based on the forces acting on the block, we can set up an inequality to find the range of angular speeds for which the block will not slip:
F_friction ≤ F_friction_max
μ * N ≤ μ_s * N
Since N = m * g * cos(Θ),
μ * m * g * cos(Θ) ≤ μ_s * m * g * cos(Θ)
Canceling out the common terms, we get:
μ ≤ μ_s
Therefore, the range of angular speeds for which the block will not slip is unrestricted. As long as the coefficient of friction between the block and the bowl surface (μ) is less than or equal to the coefficient of static friction (μ_s), the block will not slip.