A particle is held at a point on a smooth solid hemisphere of radius 3 m and centre . The particle is released and slides down the hemisphere under gravity before leaving the hemisphere at point . Given that makes an angle of with the upward vertical, find the angle that makes with the upward vertical. Take and give your answer correct to one decimal place.
A bead of mass 0.9 kg is attached to one end of a light rod of length 0.3 m. The other end of the rod is fixed at a point , about which the rod can freely rotate in a vertical plane. The bead was at rest vertically below when it was pushed horizontally at a speed of 14 m/s. Taking , find the bead’s speed when it passed the point vertically above . Give your answer in metres per second correct to one decimal place.
A rough horizontal disc is rotating at a constant angular speed of 7 rad/s about a vertical axis through its centre. A rock resting on this disc is on the point of slipping. Taking , determine the coefficient of friction between the rock and the disc given that the rock lies 11 cm away from the centre of the disc.
A particle resting at the highest point of a smooth sphere is barely pushed such that it slides down the sphere’s surface, moving at a speed of , where is the acceleration due to gravity and is the radius of the sphere. Determine the vertical distance travelled by the particle before it left the sphere’s surface.
A smooth sphere of radius 5.1 m is fixed to a horizontal surface. A particle of mass 2 kg at the highest point of the sphere slides down its surface from rest, leaves the sphere, and finally hits the horizontal surface. Taking , find the velocity of the particle when it hits the surface. Express the magnitude of the velocity, , in metres per second correct to one decimal place, and the direction of the velocity, , as the angle made with the horizontal to the nearest degree.
A turn on a racing track follows a horizontal circular arc of radius 74 m. The track at this turn is banked at an angle of to help cars go round it at speed without slipping. Given that the coefficient of friction between a car’s tyres and the track is 0.8 and taking , find the maximum speed the car can go around this turn without slipping. Give your answer in metres per second correct to one decimal place.
A rough horizontal disc is rotating about a vertical axis through its centre. A stone resting on this disc, at a distance of 0.3 m from its centre, is on the point of slipping. Given that the coefficient of friction between the stone and the disc is 0.1 and taking , find the angular speed of the disc in radians per second, giving your answer correct to one decimal place.