A ball is released from rest from point P of a smooth semi-spherical vessel as shown in figure. The ratio of the centripetal force and normal reaction on the ball at point Q is A while angular position of point Q is α with respect to point P. Which of the following graphs represent the correct relation between A and α when ball goes from Q to R?
To analyze the problem, we need to understand the motion of the ball in the semi-spherical vessel and how forces act on it.
Concepts Involved:
When the ball is released from rest at point \( P \), it rolls down under the influence of gravity.
The centripetal force \( F_c \) required to keep the ball moving in a circle at point \( Q \) is provided by the component of the gravitational force.
The normal reaction \( N \) acts perpendicular to the surface at point \( Q \).
Forces at Point \( Q \):
The centripetal force is given by \( F_c = m \cdot v^2 / r \), where \( v \) is the speed at \( Q \) and \( r \) is the radius of the vessel.
The normal reaction \( N \) balances the radial component of gravitational force and provides the required centripetal force, so \( N = m \cdot g \cdot \cos(\alpha) - F_c \).
Energy Conservation:
Using energy conservation between point \( P \) and \( Q \):
m \cdot g \cdot h = \frac{1}{2} \cdot m \cdot v^2
where \( h = r(1 - \cos(\alpha)) \) is the vertical height.
v^2 = 2 \cdot g \cdot r \cdot (1 - \cos(\alpha))
Ratio of Forces:
The ratio \( A \) of centripetal force \( F_c \) to normal reaction \( N \) is:
A = \frac{F_c}{N} = \frac{m \cdot v^2 / r}{m \cdot g \cdot \cos(\alpha) - m \cdot v^2 / r}
The graph of \( A \) versus \( \alpha \) will involve looking at this relationship as \( \alpha \) changes from 0 to \(\frac{\pi}{2}\).
The options given suggest that the correct graph is the one where \( A \) varies with \( \alpha \) reaching a maximum and then decreasing. The logical interpretation from the formula indicates such a trend.
