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Worksheet: Circular Motion

Q1:

A particle is held at a point 𝐴 on a smooth solid hemisphere of radius 3 m and centre 𝑂 . The particle is released and slides down the hemisphere under gravity before leaving the hemisphere at point 𝐡 . Given that 𝑂 𝐴 makes an angle of 2 6 ∘ with the upward vertical, find the angle that 𝑂 𝐡 makes with the upward vertical. Take 𝑔 = 9 . 8 / m s 2 and give your answer correct to one decimal place.

Q2:

A bead 𝐡 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 𝑂 , about which the rod can freely rotate in a vertical plane. The bead was at rest vertically below 𝑂 when it was pushed horizontally at a speed of 14 m/s. Taking 𝑔 = 9 . 8 / m s 2 , find the bead’s speed when it passed the point vertically above 𝑂 . Give your answer in metres per second correct to one decimal place.

Q3:

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 𝑔 = 9 . 8 / m s 2 , 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.

Q4:

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 √ 1 1 𝑔 π‘Ÿ 4 , where 𝑔 is the acceleration due to gravity and π‘Ÿ is the radius of the sphere. Determine the vertical distance travelled by the particle before it left the sphere’s surface.

  • A 5 π‘Ÿ 3 2
  • B 1 1 π‘Ÿ 3 2
  • C 1 1 π‘Ÿ 2 4
  • D 5 π‘Ÿ 1 6
  • E 3 3 π‘Ÿ 6 4

Q5:

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 𝑔 = 9 . 8 / m s 2 , find the velocity of the particle when it hits the surface. Express the magnitude of the velocity, 𝑣 , in metres per second correct to one decimal place, and the direction of the velocity, πœƒ , as the angle made with the horizontal to the nearest degree.

  • A 𝑣 = 1 4 . 1 / m s , πœƒ = 1 6 ∘
  • B 𝑣 = 5 . 8 / m s , πœƒ = 1 6 ∘
  • C 𝑣 = 1 3 . 6 / m s , πœƒ = 1 6 ∘
  • D 𝑣 = 1 4 . 1 / m s , πœƒ = 7 4 ∘
  • E 𝑣 = 1 3 . 6 / m s , πœƒ = 7 4 ∘

Q6:

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 1 9 ∘ 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 𝑔 = 9 . 8 / m s 2 , 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.

Q7:

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 𝑔 = 9 . 8 / m s 2 , find the angular speed of the disc in radians per second, giving your answer correct to one decimal place.