Worksheet: Uniform Circular Motion

In this worksheet, we will practice solving problems about a particle moving in a circular motion with constant velocity under constant centripetal force.

Q1:

A clockmaker is attaching an hour hand to a clock that is hanging on a vertical wall. The hour hand has a mass of π‘š and a length of 3𝑙 and it is attached to the clock through a point that is at a distance of 𝑙 from one of the ends. The clockmaker is attaching the hand so that it points at 12 o’clock. As he is trying to fix it in place, he slightly disturbs it from its position of equilibrium and it starts to rotate. Assuming that the hour hand is uniform and the axle is smooth, find the angular speed of the hour hand after it has turned through an angle of πœƒ. Give your answer in terms of 𝑙, πœƒ, and the acceleration due to gravity 𝑔.

  • A ο„ž 𝑔 2 𝑙 ( 1 βˆ’ πœƒ ) c o s
  • B ο„ž 𝑔 𝑙 ( 1 βˆ’ πœƒ ) c o s
  • C ο„ž 4 𝑔 3 𝑙 ( 1 βˆ’ πœƒ ) c o s
  • D ο„ž 4 𝑔 3 𝑙 ( 1 βˆ’ πœƒ ) s i n
  • E ο„ž 𝑔 𝑙 ( 1 βˆ’ πœƒ ) s i n

Q2:

An airplane needed to change direction from a bearing of 005∘ to a bearing of 050∘. It did this by banking at an angle of 𝛼 to the horizontal. The banking maneuver caused the airplane to travel in a horizontal circular arc so that, after 30 seconds, it was pointing in the right direction. Given that the plane’s speed was 359 km/h throughout the maneuver, find the two possible values of 𝛼. Give your answers correct to one decimal place. Take 𝑔=9.8/ms.

  • A 6 7 . 7 ∘ , 8 6 . 6 ∘
  • B 6 1 . 8 ∘ , 6 7 . 7 ∘
  • C 6 7 . 7 ∘ , 6 1 . 8 ∘
  • D 6 1 . 8 ∘ , 1 4 . 9 ∘
  • E 6 7 . 7 ∘ , 1 4 . 9 ∘

Q3:

A particle 𝑃 of mass π‘š lies on a rotating rough disk 5π‘Ž4 away from its center, where the coefficient of friction between the particle and the disk is 12. Given that the disk is rotating horizontally at a constant angular speed πœ” about its vertical axis and that 𝑃 remains at rest relative to the disk when πœ”β‰€π‘₯, find π‘₯. Consider the acceleration due to gravity to be 𝑔.

  • A 𝑔 5 π‘Ž
  • B 2 𝑔 1 5 π‘Ž
  • C 3 𝑔 5 π‘Ž
  • D 4 𝑔 5 π‘Ž
  • E 2 𝑔 5 π‘Ž

Q4:

On his way home, Mason came across a traffic circle of radius 18 m while driving in his car. Suppose he drove round the traffic circle at 26 km/h, determine his acceleration correct to two decimal places.

Q5:

A bend in a road follows a horizontal circular arc of radius 107 m. At what angle to the horizontal should the road be banked so that a car traveling at 16 km/h experiences no frictional force perpendicular to its direction of motion? Give your answer correct to one decimal place. Take 𝑔=9.8/ms.

Q6:

A bead of mass 55 g was threaded on a wire. The wire was bent into a circular hoop of radius 0.99 m. The bead moves around the hoop at a constant speed of 4.5 m/s. Given that the hoop lies in a horizontal plane, find the vertical component 𝐹 and the horizontal component 𝐹 of the reaction force acting on the bead due to the wire. Take 𝑔=9.8/ms.

  • A 𝐹 = 1 . 1 2 5  N , 𝐹 = 0 . 5 3 9  N
  • B 𝐹 = 0 . 5 3 9  N , 𝐹 = 1 , 1 2 5  N
  • C 𝐹 = 5 3 9  N , 𝐹 = 1 . 1 2 5  N
  • D 𝐹 = 0 . 5 3 9  N , 𝐹 = 1 . 1 2 5  N
  • E 𝐹 = 0 . 5 3 9  N , 𝐹 = 1 . 1 0 3  N

Q7:

An athlete goes around a circular track at 3.7 m/s. If the radius of the track is 19 m, how long does one lap take? Round your answer to one decimal place.

Q8:

A bead of mass 155 g is attached by a light inextensible string of length 31 cm to a fixed point 𝑂 on the smooth horizontal surface of a table. What is the tension in the string if the bead moves at a constant speed of 14 m/s around 𝑂 with the string taut at all times?

Q9:

A light inextensible string 𝐴𝐡 is 23 cm long and passes through a small smooth hole in the centre of a smooth horizontal surface. The string is attached to two particles 𝐴 and 𝐡, one at either end of the string. Both particles have mass 60 g. Particle 𝐴 lies on the smooth horizontal surface, while particle 𝐡 hangs freely below. Given that 𝐴 is moving in a circular path about the hole at a constant linear speed, determine how fast its linear speed should be so that 𝐡 stays in equilibrium 18 cm below the surface. Consider the acceleration due to gravity to be 9.8 m/s2.

  • A 7 1 0 m/s
  • B 2 1 √ 1 0 5 0 m/s
  • C7 m/s
  • D14 m/s
  • E 7 5 m/s

Q10:

Find the acceleration of a particle moving in a circle of radius 11 cm at a constant angular speed of 2 rad/s.

Q11:

Find the centripetal force experienced by a particle of mass 250 g which moves on a circle of radius 20 cm at a constant angular speed of 4 rad/s.

Q12:

Find the centripetal force experienced by a particle of mass 50 g which moves on a circle of radius 14 cm at a constant speed of 28 cm/s.

Q13:

A pebble of mass 21 g rests on a wooden merry-go-round 12 cm away from its center. Suppose the merry-go-round rotates at a constant angular speed of 3 rad/s and the pebble does not slip. Determine, in dynes, the force of friction acting on the pebble.

Q14:

A ball of mass 3 kg is attached to one end of a light inextensible string of length 45 cm. The other end of the string is fixed at a point 𝐴. The ball moves at a constant angular speed of πœ” in a horizontal circle of radius 37 cm whose center is vertically below 𝐴. Taking 𝑔=9.8/ms, determine the tension 𝑇 in the string, giving your answer in newtons correct to one decimal place. Hence find the value of πœ” in radians per second correct to one decimal place.

  • A 𝑇 = 5 1 . 7 N , πœ” = 5 . 1 / r a d s
  • B 𝑇 = 3 5 . 8 N , πœ” = 6 . 2 / r a d s
  • C 𝑇 = 3 5 . 8 N , πœ” = 5 . 1 / r a d s
  • D 𝑇 = 5 1 . 7 N , πœ” = 0 . 6 / r a d s
  • E 𝑇 = 5 1 . 7 N , πœ” = 6 . 2 / r a d s

Q15:

A marble of mass 52 g hangs from the ceiling by a light inextensible string 0.82 m long. Given that the string is fixed to the ceiling at point 𝐴 and that the marble moves in a horizontal circular path whose center is 0.65 m vertically below 𝐴 at a constant angular speed, determine, to the nearest decimal place, the tension in the string 𝑇 and the angular speed of the marble πœ”. Take 𝑔=9.8/ms.

  • A 𝑇 = 0 . 8 N , πœ” = 4 . 4 / r a d s
  • B 𝑇 = 0 . 4 N , πœ” = 3 . 5 / r a d s
  • C 𝑇 = 0 . 4 N , πœ” = 3 . 9 / r a d s
  • D 𝑇 = 0 . 6 N , πœ” = 3 . 5 / r a d s
  • E 𝑇 = 0 . 6 N , πœ” = 3 . 9 / r a d s

Q16:

A bead 𝐡 of mass 2.7 kg is attached to one end of a light rod of length 1.6 m. The other end of the rod is fixed at a point 𝑂, about which the rod freely rotates in a vertical plane. When the bead is vertically below 𝑂, its speed is 8.5 m/s. Taking 𝑔=9.8/ms, find the smaller angle between 𝑂𝐡 and the vertical when the tension in the rod is zero. Give your answer correct to two decimal places.

Q17:

A particle of mass 26 g lies on the inner surface of a vertical hollow cylinder of radius 83 cm. The cylinder rotates about its axis of symmetry at 24 rev/s. What is the normal reaction between the cylinder and the particle? Give your answer in newtons correct to one decimal place.

Q18:

A smooth solid hemisphere of radius π‘Ÿ rests with its plane face on a horizontal surface. A particle resting at the highest point of the hemisphere is given initial speed ο„žπ‘”π‘Ÿ7, where 𝑔 is the acceleration due to gravity. The particle slides down the sphere, leaves its surface, and finally hits the horizontal surface at the point 𝐴. Find the particle’s speed at 𝐴.

  • A 4 7 √ 7 𝑔 π‘Ÿ
  • B √ 1 0 5 𝑔 π‘Ÿ 7
  • C 2 7 √ 1 4 𝑔 π‘Ÿ
  • D √ 9 1 𝑔 π‘Ÿ 7
  • E √ 2 1 0 𝑔 π‘Ÿ 1 4

Q19:

A smooth solid hemisphere with a radius of 10 m and centre 𝑂 rests with its flat face on a horizontal surface. Suppose that a ball 𝐡 with mass 7 kg resting on top of the hemisphere’s highest point is slightly pushed. Find the angle between 𝑂𝐡 and the upward vertical when the ball leaves the surface of the hemisphere. Give your answer correct to one decimal place.

Q20:

A particle 𝑃 is placed at the highest point of a smooth solid hemisphere of radius π‘Ÿ and center 𝑂. The particle is projected horizontally from this point with a speed of ο„žπ‘Ÿπ‘”4, where 𝑔 is the acceleration due to gravity. Let 𝑣 be the particle’s speed when 𝑂𝑃 makes an angle of πœƒ with the upward vertical as the particle slides down the hemisphere. Find an expression for π‘£οŠ¨ in terms of π‘Ÿ, 𝑔, and πœƒ.

  • A 9 𝑔 π‘Ÿ 4 βˆ’ 2 𝑔 π‘Ÿ πœƒ s i n
  • B 9 𝑔 π‘Ÿ 4 βˆ’ 2 𝑔 π‘Ÿ πœƒ c o s
  • C 3 𝑔 π‘Ÿ 2 βˆ’ 𝑔 π‘Ÿ πœƒ c o s
  • D 9 𝑔 π‘Ÿ 4 + 2 𝑔 π‘Ÿ πœƒ c o s
  • E 3 𝑔 π‘Ÿ 2 + 𝑔 π‘Ÿ πœƒ c o s

Q21:

A ball of mass 4.7 kg hangs from the ceiling by a light inextensible string that is 6 m long. Given that the string is fixed to the ceiling at point 𝐴 and that the ball moves in a horizontal circular path whose center is vertically below 𝐴 at a constant angular speed such that it completes one revolution in 1.4 seconds, find the angle the string makes with the vertical giving your answer to the nearest degree. Take 𝑔=9.8/ms.

Q22:

A smooth solid hemisphere has its flat surface lying on a horizontal table and its curved surface facing upwards. The flat face of the hemisphere has centre 𝑂 and radius π‘Ÿ. Point 𝐴, at which a particle 𝑃 is placed, is the highest point on the hemisphere. 𝑃 is then given an initial horizontal speed 𝑒, where 𝑒=π‘”π‘Ÿ7 and the acceleration due to gravity is 𝑔=9.8/ms. Find the angle between the velocity of 𝑃 and the table as 𝑃 strikes the table, giving your answer to the nearest degree.

Q23:

A bead 𝐡 of mass 1 kg is attached to one end of a light rod of length 2 m. The other end of the rod is fixed at a point 𝑂 about which the rod freely rotates in a vertical plane. When the bead is vertically below 𝑂, its speed is 3.5 m/s. Taking 𝑔=9.8/ms, find the bead’s speed when the tension in the rod is zero. Give your answer in meters per second correct to one decimal place.

Q24:

A particle of mass 0.7 kg is attached to end 𝐴 of a light rod 𝐴𝐡 of length 0.7 m. The rod is free to rotate in a vertical plane about 𝐡. The particle is held at rest with 𝐴𝐡 at 45∘ to the upward vertical. The particle is released. Calculate the tension in the rod as the particle passes through the lowest point of the path, giving your answer to the nearest two decimal places. Consider the acceleration due to gravity to be 9.8 m/s2.

Q25:

A smooth solid hemisphere has its flat surface lying on a horizontal table and its curved surface facing upwards. The flat face of the hemisphere has centre 𝑂 and radius π‘Ÿ. Point 𝐴, at which a particle 𝑃 is placed, is the highest point on the hemisphere. 𝑃 is then given an initial horizontal speed 𝑒, where 𝑒=π‘”π‘Ÿ6 and the acceleration due to gravity is 𝑔=9.8/ms. Find the angle at which 𝑃 strikes the table to the nearest degree.

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