# Advanced Higher Physics. Rotational motion

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1 Wallace Hall Academy Physics Department Advanced Higher Physics Rotational motion Problems AH Physics: Rotational Motion

2 Data Common Physical Quantities QUANTITY SYMBOL VALUE Gravitational acceleration g 9.8 m s -2 Radius of Earth R E 6.4 x 10 6 m Mass of Earth M E 6.0 x kg Mass of Moon M M 7.3 x kg Mean radius of Moon orbit 3.84 x 10 8 m Universal constant of gravitation G 6.67 x m 3 kg -1 s -2 Speed of light in vacuum c 3.0 x 10 8 m s -1 Speed of sound in air v 3.4 x 10 2 m s -1 Mass of electron m e 9.11 x kg Charge on electron e x C Mass of neutron m n x kg Mass of proton m p x kg Planck s constant h 6.63 x J s Permittivity of free space x F m -1 Permeability of free space 0 4 x 10-7 H m -1 Astronomical Data Planet or satellite Mass/ kg Density/ kg m -3 Radius/ m Grav. accel./ m s -2 Escape velocity/ m s -1 Mean dist from Sun/ m Mean dist from Earth/ m Sun 1.99x x x x x Earth 6.0 x x x x x Moon 7.3 x x x x x 10 8 Mars 6.4 x x x x x Venus 4.9 x x x x x AH Physics: Rotational Motion

3 TUTORIAL 1.0 Equations of motion 1 The displacement, s in metres, of an object after a time, t in seconds, is given by s = 90t 4 t 2 (a) Find by differentiation the equation for its velocity. (b) At what time will the velocity be zero? (c) Show that the acceleration is a constant and state its value. 2 Given that a = dv, show by integration that the velocity, v, is given by dt v = u + at. State clearly the meaning of the symbol, u, in this equation. 3. Given that v = ds and v = u + at, show by integration that dt s = ut + ½ at 2. Where the symbols have their usual meaning. 4. The displacement, s, of a moving object after a time, t, is given by s = 8 10t + t 2. Show that the unbalanced force acting on the object is constant. 5. The displacement, s, of an object after time, t, is given by s = 3t 3 + 5t. (a) Derive an expression for the acceleration of the object. (b) Explain why this expression indicates that the acceleration is not constant. 6. A trolley is released from the top of a runway which is 6 m long. The displacement, s in metres, of the trolley is given by the expression s = 5t + t 2, where t is in seconds. Determine: (a) an expression for the velocity of the trolley (b) the acceleration of the trolley (c) the time it takes the trolley to reach the bottom of the runway (d) the velocity of the trolley at the bottom of the runway. 7. A box slides down a smooth slope with an acceleration of 4 m s -2. The velocity of the box at a time t = 0 is 3 m s -1 down the slope. Using a = dv show by integration that the velocity, v, of the box is given by dt v = 3 + 4t. 8. The equation for the velocity, v, of a moving trolley is v = 2 + 6t. Using v = ds derive an expression for the displacement, s, of the trolley. dt 9 A projectile is launched from the top of a building with an initial speed of 20 m s -1 at an angle of 30 to the horizontal. The height of the building is 30 m. (a) Calculate how long it takes the projectile to reach the ground. (b) Calculate the velocity of the projectile on impact with the ground, (magnitude and direction). AH Physics: Rotational Motion

5 TUTORIAL 2.1 Angular Motion 1 If 2 radians equals 360, calculate the number of degrees in one radian. 2 Calculate the angular velocity in rad s -1 of the second hand of an analogue watch. 3 The graph below shows the variation of angular velocity with time for a rotating body. /rad s -1 t /s (a) (b) (c) Find the angular displacement covered in the first 3 seconds. Find the total angular displacement for the 6 seconds. Calculate the angular acceleration of the rotating body. 4 A wheel accelerates uniformly from rest. After 12 s the wheel is completing 100 revolutions per minute (r.p.m.) (a) Convert 100 r.p.m. to its equivalent value in rad s -1. (b) Calculate the average angular acceleration of the wheel. 5 The angular velocity of a car engine s drive shaft is increased from 100 rad s -1 to 300 rad s -1 in 10 s. (a) (b) Calculate the angular acceleration of the drive shaft. Calculate the angular displacement during this time. (c) A point on the rim of the drive shaft is at a radius of 0.12 m. Calculate the distance covered by this point in the 10 s time interval. 6 Use calculus methods to derive the equations for angular motion. The method is very similar to that for linear motion. Note: in the unit or course assessment you may be asked to derive the linear motion equations but not the angular motion equations. AH Physics: Rotational Motion

7 TUTORIAL 3.1 Circular Motion 1 An Earth satellite is required to be in a circular orbit at a distance of 7.5 x 10 6 m from the centre of the Earth. The central force is due to the gravitational force. The acceleration due to the Earth s gravity at this point is 7.0 m s -2 Find: (a) the required satellite speed (b) the period of revolution of the satellite. 2 What would be the period of rotation of the Earth about its axis if its speed of rotation increased to such an extent that an object at the equator became weightless? (Hint: equate mg to mv2 ). r 3 A sphere of mass 0.20 kg is rotating in a circular path at the end of a string 0.80 m long. The other end of the string is fixed. The period of the motion is 0.25 s. (a) Calculate the tension in the string, which you may assume to be horizontal. (b) In practice the string is not horizontal. Explain why this is so. (c) Draw a force diagram for the sphere. From this calculate the angle the string would make with the horizontal. 4 The moon takes 27.3 days (2.0 x 10 6 s) to complete one orbit of the Earth. The distance between the centres of the Earth and Moon is 4.0 x 10 8 m. Calculate the magnitude of the Moon s acceleration towards the Earth. 5 A ball of mass 2.0 kg is attached to a string 1.6 m long and is made to travel in a vertical circle. The ball passes its highest point with a speed of 5.0 m s -1. (a) What is the kinetic energy of the ball at its highest point? (b) What is its potential energy when it is at the highest point (with reference to its lowest point)? (c) What is its kinetic energy at the lowest point? (d) What is its speed at the lowest point? (e) What is the tension in the string at the highest and lowest points? (f) What is the least speed the ball could have at the highest point in order to be able to complete a vertical circle at all? 6 An old humpback bridge has a radius of curvature of 20 m. What is the maximum speed at which a car can pass over this bridge if the car is not to leave the road surface? 7 (a) A pail of water is swinging in a vertical circle of radius 1.2 m, so that the water does not fall out. What is the minimum linear speed required for the pail of water. (b) Convert this speed into an angular velocity. 8 An object of mass 0.20 kg is connected by a string to an object of half its mass. The smaller mass is rotating at a radius of 0.15 m on a table which has a frictionless surface. The larger mass is suspended through a hole in the middle of the table. Calculate the number of revolutions per minute the smaller mass must make so that the larger mass is stationary. AH Physics: Rotational Motion

8 Banking of a Track 9 A circular track of radius 60 m is banked at angle. A car is driven round the track at 20 m s -1. (a) Draw a diagram showing the forces acting on the car. (b) Calculate the angle of banking required so that the car can travel round the track without relying on frictional forces (i.e. no side thrust supplied by friction on the track surface). Conical Pendulum 10 A small object of mass m revolves in a horizontal circle at constant speed at the end of a string of length 1.2 m. As the object revolves, the string sweeps out the surface of a right circular cone m long m The cone has semi-angle 30. Calculate: (a) the period of the motion; (b) the speed of the object. [Hint: try resolving the tension in the string into horizontal and vertical components.] AH Physics: Rotational Motion

12 TUTORIAL 5.1 Torque, Moments of Inertia and Angular Momentum 1 A flywheel has a moment of inertia of 1.2 kg m 2. The flywheel is acted on by a torque of magnitude 0.80 N m. (a) Calculate the angular acceleration produced. (b) The torque acts for 5.0 s and the flywheel starts from rest. Calculate the angular velocity at the end of the 5.0 s. 2 A mass of 0.10 kg is hung from the axle of a flywheel as shown below. The mass is released from a height of 2.0 m above ground level. axis of rotation 0.10 kg ground 2.0 m The following results were obtained in the experiment: time for mass to fall to the ground t = 8.0 s radius of axle R = 0.10 m. (a) By energy considerations, show that the final speed of the flywheel is given by v = 392. where I is the moment of inertia of the flywheel. Friction effects are I ignored. (b) Calculate the moment of inertia of the flywheel. 3 A heavy drum has a moment of inertia of 2.0 kg m 2. It is rotating freely at 10 rev s -1 and has a radius of 0.50 m. A constant frictional force of 5.0 N is then exerted at the rim of the drum. (a) Calculate the time taken for the drum to come to rest. (b) Calculate the angular displacement in this time. (c) Hence calculate the heat generated in the braking action. AH Physics: Rotational Motion

13 4 A cycle wheel is mounted so that it can rotate horizontally as shown. Data on wheel: radius of wheel = 0.50 m, mass of wheel = 2.0 kg. thin spokes axle (a) Calculate the moment of inertia of the wheel system. State any assumptions you make. (b) A constant driving force of 20 N is applied to the rim of the wheel. (i) Calculate the magnitude of the driving torque on the wheel. (ii) Calculate the angular acceleration of the wheel. (c) After a period of 5.0 s, calculate: (i) the angular displacement, (ii) the angular momentum of the wheel, and (iii) the kinetic energy of the wheel. 5 A very light but strong disc is mounted on a free turning bearing as shown below m disc 0.20 kg A mass of 0.20 kg is placed at a radius of 0.40 m and the arrangement is set rotating at 1.0 rev s -1. (The moment of inertia of the disc can be considered to be negligible.) (a) Calculate the angular momentum of the 0.20 kg mass. (b) Calculate the kinetic energy of the mass. (c) The mass is pushed quickly into a radius of 0.20 m. By applying the principle of conservation of angular momentum, calculate the new angular velocity of the mass in rad s -1. (d) Find the new kinetic energy of the mass and account for any difference. AH Physics: Rotational Motion

14 6 A uniform metal rod has a mass, M, of 1.2 kg and a length, L, of 1.0 m. Clamped to each end of the rod is a mass of 0.50 kg as shown below. axis 10 N 0.50 kg 1.0 m (a) Calculate an approximate value for the moment of inertia of the complete arrangement about the central axis as shown. Assume that Irod = 1 12 ML2 about this axis. (b) The arrangement is set rotating by a force of 10 N as shown in the diagram. The force acts at a tangent to the radius. (i) Calculate the applied torque. (ii) Hence find the maximum angular acceleration. You may assume that the force of friction is negligible. (iii) Calculate the kinetic energy of the arrangement 4.0 s after it is set rotating. 7 An unloaded flywheel, which has a moment of inertia of 1.5 kg m 2, is driven by an electric motor. The flywheel is rotating with a constant angular velocity of 52 rad s -1. The driving torque, of 7.7 N m, supplied by the motor is now removed. How long will it take for the flywheel to come to rest. You may assume that the frictional torque remains constant? 8 A solid aluminium cylinder and a hollow steel cylinder have the same mass and radius. The two cylinders are released together at the top of a slope. (a) Which of the two cylinders will reach the bottom first? (b) Explain your answer to part (a). AH Physics: Rotational Motion

15 9 A solid cylinder and a hollow cylinder each having the same mass M and same outer radius R, are released at the same instant at the top of a slope 2.0 m long as shown below. The height of the slope is 0.04 m. 2.0 m h h = 0.04 m R r R s olid cylinder end-on view of the cylinders hollow cylinder M = 10 kg, R = 0.10 m M = 10 kg, R = 0.10 m, r = 0.05 m I = 1 2 M R2 I = 1 2 M (R2 + r 2 ) It is observed that one of the cylinders reaches the bottom of the slope before the other. (a) Using the expressions given above, show that the moments of inertia for the cylinders are as follows: (i) solid cylinder; I = 0.05 kg m 2 (ii) hollow cylinder; I = kg m 2. (b) By energy considerations, show that the linear velocity of any cylinder at the bottom of the slope is given by: v = 2 g h. I [1 + M R 2] (c) Using the expression in (b) above, calculate the velocities of the two cylinders at the bottom of the slope and hence show that one of the cylinders arrives at the bottom of the slope 0.23 s ahead of the other. AH Physics: Rotational Motion

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