12. Motion in a circle
A section of Physics, 9702
Listing 10 of 15 questions
Define the radian. A small mass is attached to a string. The mass is rotating about a fixed point P at constant speed, as shown in . mass rotating at constant speed P Explain what is meant by the angular speed about point P of the mass. A horizontal flat plate is free to rotate about a vertical axis through its centre, as shown in . M d plate A small mass M is placed on the plate, a distance d from the axis of rotation. The speed of rotation of the plate is gradually increased from zero until the mass is seen to slide off the plate. The maximum frictional force F between the plate and the mass is given by the expression F = 0.72W, where W is the weight of the mass M. The distance d is 35 cm. Determine the maximum number of revolutions of the plate per minute for the mass M to remain on the plate. Explain your working. number = The plate in is covered, when stationary, with mud. Suggest and explain whether mud near the edge of the plate or near the centre will first leave the plate as the angular speed of the plate is slowly increased.
9702_s08_qp_4
THEORY
2008
Paper 4, Variant 0
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Define the radian. A stone of weight 3.0 N is fixed, using glue, to one end P of a rigid rod CP, as shown in . 85 cm C P ω glue stone, weight 3.0 N The rod is rotated about end C so that the stone moves in a vertical circle of radius 85 cm. The angular speed ω of the rod and stone is gradually increased from zero until the glue snaps. The glue fixing the stone snaps when the tension in it is 18 N. For the position of the stone at which the glue snaps, on the dotted circle of , mark with the letter S the position of the stone, calculate the angular speed ω of the stone. angular speed = rad s–1
9702_s10_qp_41
THEORY
2010
Paper 4, Variant 1
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Define the radian. A circular metal disc spins horizontally about a vertical axis, as shown in . 9.3 cm 1.2 cm rotation axis modelling clay metal disc (not to scale) A piece of modelling clay is attached to the disc. For the instant when the piece of modelling clay is in the position shown, draw on : an arrow, labelled V, showing the direction of the velocity of the modelling clay an arrow, labelled A, showing the direction of the acceleration of the modelling clay. The metal disc in has a radius of 9.3 cm. The centre of gravity of the modelling clay is 1.2 cm from the rim of the disc and moves with a speed of 0.68 m s–1. Calculate the angular speed ω of the disc. ω = rad s–1 Calculate the acceleration a of the centre of gravity of the modelling clay. a = m s–2 A second piece of modelling clay is attached to the disc in the position shown in . first piece of modelling clay second piece of modelling clay The second piece of modelling clay has a larger mass than the first piece. By placing one tick (3) in each row, complete Table 1.1 to show how the quantities indicated compare for the two pieces of modelling clay. Table 1.1 quantity less for second piece than first piece same for both pieces greater for second piece than first piece angular speed linear speed acceleration , ,
9702_s24_qp_42
THEORY
2024
Paper 4, Variant 2
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Explain what is meant by a radian, why one complete revolution is equivalent to an angular displacement of 2π rad. An elastic cord has an unextended length of 13.0 cm. One end of the cord is attached to a fixed point C. A small mass of weight 5.0 N is hung from the free end of the cord. The cord extends to a length of 14.8 cm, as shown in . 14.8 cm C small mass The cord and mass are now made to rotate at constant angular speed ω in a vertical plane about point C. When the cord is vertical and above C, its length is the unextended length of 13.0 cm, as shown in . Examiner’s Use 13.0 cm C L C Show that the angular speed ω of the cord and mass is 8.7 rad s–1. The cord and mass rotate so that the cord is vertically below C, as shown in . Calculate the length L of the cord, assuming it obeys Hooke’s law. L = cm
9702_w07_qp_4
THEORY
2007
Paper 4, Variant 0
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A planet of mass m is in a circular orbit of radius r about the Sun of mass M, as illustrated in . Sun mass M planet r mass m The magnitude of the angular velocity and the period of revolution of the planet about the Sun are x and T respectively. State what is meant by angular velocity, the relation between x and T. Show that, for a planet in a circular orbit of radius r, the period T of the orbit is given by the expression T 2 = cr 3 where c is a constant. Explain your working. Data for the planets Venus and Neptune are given in . planet r / 108 km T / years Venus Neptune 1.08 45.0 0.615 Assume that the orbits of both planets are circular. Use the expression in to calculate the value of T for Neptune. T = years Determine the linear speed of Venus in its orbit. speed = km s–1
9702_w10_qp_43
THEORY
2010
Paper 4, Variant 3
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With reference to velocity and acceleration, describe uniform circular motion. Two cars are moving around a horizontal circular track. One car follows path X and the other follows path Y, as shown in . start and finish line track path X path Y 318 m 27 m (not to scale) The radius of path X is 318 m. Path Y is parallel to, and 27 m outside, path X. Both cars have mass 790 kg. The maximum lateral friction force F that the cars can experience without sliding is the same for both cars. The maximum speed at which the car on path X can move around the track without sliding is 94 m s–1. Calculate F. F = N Both cars move around the track. Each car has the maximum speed at which it can move without sliding. Complete Table 1.1, by placing one tick in each row, to indicate how the quantities indicated for the car on path Y compare with the car on path X. Table 1.1 Y less than X Y same as X Y greater than X centripetal acceleration maximum speed time taken for one lap of the track
9702_w21_qp_41
THEORY
2021
Paper 4, Variant 1
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With reference to velocity and acceleration, describe uniform circular motion. Two cars are moving around a horizontal circular track. One car follows path X and the other follows path Y, as shown in . start and finish line track path X path Y 318 m 27 m (not to scale) The radius of path X is 318 m. Path Y is parallel to, and 27 m outside, path X. Both cars have mass 790 kg. The maximum lateral friction force F that the cars can experience without sliding is the same for both cars. The maximum speed at which the car on path X can move around the track without sliding is 94 m s–1. Calculate F. F = N Both cars move around the track. Each car has the maximum speed at which it can move without sliding. Complete Table 1.1, by placing one tick in each row, to indicate how the quantities indicated for the car on path Y compare with the car on path X. Table 1.1 Y less than X Y same as X Y greater than X centripetal acceleration maximum speed time taken for one lap of the track
9702_w21_qp_43
THEORY
2021
Paper 4, Variant 3
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Define the radian. The minute hand of a clock revolves at constant angular speed around the face of the clock, completing one revolution every hour. A small piece of modelling clay is attached to the hand with its centre of gravity at a distance L from the fixed end of the hand, as shown in . L free end face of clock modelling clay direction of revolution of minute hand minute hand fixed end Calculate the angular speed ω of the minute hand. ω = rad s–1 During a time interval of 1400 s, the centre of gravity of the piece of modelling clay in moves through a total distance of 0.44 m. Calculate the angle through which the minute hand moves in this time interval. angle = rad Determine distance L. L = m Calculate the magnitude of the centripetal acceleration of the piece of modelling clay. centripetal acceleration = m s–2 Use your answer in to explain why the variation with time of the magnitude of the force exerted by the minute hand on the piece of modelling clay is negligible as the minute hand undergoes one full revolution.
9702_w23_qp_42
THEORY
2023
Paper 4, Variant 2
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A large bowl is made from part of a hollow sphere. A small spherical ball is placed inside the bowl and is given a horizontal speed. The ball follows a horizontal circular path of constant radius, as shown in . 14 cm ball The forces acting on the ball are its weight W and the normal reaction force R of the bowl on the ball, as shown in . ball wall of bowl W R e The normal reaction force R is at an angle θ to the horizontal. By resolving the reaction force R into two perpendicular components, show that the resultant force F acting on the ball is given by the expression W = F tan θ. State the significance of the force F for the motion of the ball in the bowl. The ball moves in a circular path of radius 14 cm. For this radius, the angle θ is 28°. Calculate the speed of the ball. speed = m s−1
9702_w14_qp_41
THEORY
2014
Paper 4, Variant 1
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A large bowl is made from part of a hollow sphere. A small spherical ball is placed inside the bowl and is given a horizontal speed. The ball follows a horizontal circular path of constant radius, as shown in . 14 cm ball The forces acting on the ball are its weight W and the normal reaction force R of the bowl on the ball, as shown in . ball wall of bowl W R e The normal reaction force R is at an angle θ to the horizontal. By resolving the reaction force R into two perpendicular components, show that the resultant force F acting on the ball is given by the expression W = F tan θ. State the significance of the force F for the motion of the ball in the bowl. The ball moves in a circular path of radius 14 cm. For this radius, the angle θ is 28°. Calculate the speed of the ball. speed = m s−1
9702_w14_qp_42
THEORY
2014
Paper 4, Variant 2
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Questions Discovered
15