Lesson Worksheet: Uniform Circular Motion Mathematics

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.

  • Aο„žπ‘”2𝑙(1βˆ’πœƒ)cos
  • Bο„žπ‘”π‘™(1βˆ’πœƒ)cos
  • Cο„ž4𝑔3𝑙(1βˆ’πœƒ)cos
  • Dο„ž4𝑔3𝑙(1βˆ’πœƒ)sin
  • Eο„žπ‘”π‘™(1βˆ’πœƒ)sin

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.

  • A67.7∘, 86.6∘
  • B61.8∘, 67.7∘
  • C67.7∘, 61.8∘
  • D61.8∘, 14.9∘
  • E67.7∘, 14.9∘

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.

  • A𝑔5π‘Ž
  • B2𝑔15π‘Ž
  • C3𝑔5π‘Ž
  • D4𝑔5π‘Ž
  • E2𝑔5π‘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 is threaded on a wire. The wire is 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.

  • A𝐹=1.125N, 𝐹=0.539N
  • B𝐹=0.539N, 𝐹=1,125N
  • C𝐹=539N, 𝐹=1.125N
  • D𝐹=0.539N, 𝐹=1.125N
  • E𝐹=0.539N, 𝐹=1.103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.

  • A710 m/s
  • B21√1050 m/s
  • C7 m/s
  • D14 m/s
  • E75 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.

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