Friction

A football of radius is kept on a hole of radius made on a plank kept horizontally. One end of the plank is now lifted so that it gets tilted making an angle from the horizontal as shown in the figure below. The maximum value of so that the football does not start rolling down the plank satisfies (figure is schematic and not drawn to scale)

Put a uniform meter scale horizontally on your extended index fingers with the left one at and the right one at .When you attempt to move both the fingers slowly towards the center, initially only the left finger slips with respect to the scale and the right finger does not. After some distance, the left finger stops and the right one starts slipping. Then the right finger stops at a distance from the center of the scale and the left one starts slipping again. This happens because of the difference in the frictional forces on the two fingers. If the coefficients of static and dynamic friction between the fingers and the scale are and , respectively, the value of (in ) is____.

A slide with a frictionless curved surface, which becomes horizontal at its lower end, is fixed on the terrace of a building of height 3ℎ from the ground, as shown in the figure. A spherical ball of mass 𝑚 is released on the slide from rest at a height ℎ from the top of the terrace. The ball leaves the slide with a velocity and falls on the ground at a distance 𝑑 from the building making an angle θ with the horizontal. It bounces off with a velocity of v and reaches a maximum height of ℎ₁. The acceleration due to gravity is 𝑔 and the coefficient of restitution of the ground is . Which of the following statement(s) is(are) correct?

A thin uniform rod of length L and certain mass is kept on a frictionless horizontal table with a massless string of length L fixed to one end (top view is shown in the figure). The other end of the string is pivoted to a point O. If a horizontal impulse P is imparted to the rod at a distance x = L/n from the mid-point of the rod (see figure), then the rod and string revolve together around the point O, with the rod remaining aligned with the string. In such a case, the value of n is _____.

A thin uniform rod of length L and certain mass is kept on a frictionless horizontal table with a massless string of length L fixed to one end (top view is shown in the figure). The other end of the string is pivoted to a point O. If a horizontal impulse P is imparted to the rod at a distance x = L/n from the mid-point of the rod (see figure), then the rod and string revolve together around the point O, with the rod remaining aligned with the string. In such a case, the value of n is _____.