ASRJC Motion in a Circle Notes

ANDERSON SERANGOON JUNIOR COLLEGE PHYSICS 9749 6-1 Additional Notes: Topic 6: Motion in a Circle Content: • Kinematics of uniform circular motion • Centripetal acceleration • Centripetal force Learning Outcomes: Candidates should be able to: Kinematics of uniform circular motion (a) express angular displacement in radians (b) show an understanding of and use the concept of angular velocity to solve problems. (c) recall and use rv = to solve problems. Centripetal acceleration (d) describe qualitatively motion in a curved path due to a perpendicular force, and understand the centripetal acceleration in the case of uniform motion in a circle. (e) recall and use centripetal acceleration 2ra = and r va 2 = to solve problems. Centripetal force (f) recall and use centripetal force 2mrF = and r mvF 2 = to solve problems. Demonstrating Science Inquiry Skills In this chapter, we tackle key questions such as, “What causes an object to move in a circular path?”, “What is a centripetal force?” Relating Science and Society An example of an application of the concept of motion in a circle is the determination of the speed limit of vehicles negotiating a circular path. Other examples of motion in a circle include: • the rotation of the Earth about its axis, • the motion of the moon about the Earth (approximately circular), • the rotation of a compact disc about its axis, • the motion of a Ferris wheel, • the motion of the second, minute and hour hands of an analogue clock. Watch the human loop the loop on YouTube: https://youtu.be/ GkPudmBQBEY

ANDERSON SERANGOON JUNIOR COLLEGE PHYSICS 9749 6-2 Additional Notes: 6A Kinematics of Uniform Circular Motion 6A.1 Angular Displacement  Consider an object going round a circular path of radius r from point A to B. The angular displacement  is the angle the radius sweeps through in a specific direction from a reference line. s r=θ Note: • Since  is a ratio of a length to another length, it is dimensionless. •  is a vector quantity since it has direction (clockwise / anti-clockwise). Check Your Understanding 1 How many radians does 1° correspond to? A 0.035 rad B 0.0087 rad C 0.017 rad Recall that  rad = 180 Hence, =1 rad 180  r s A B O Definition of angular displacement: The angle through which an object turns, usually measured in radians (rad). Definition of a radian: 1 radian is the angle subtended by an arc of equal length to the radius. Legend: s: arc length r: radius of circle

ANDERSON SERANGOON JUNIOR COLLEGE PHYSICS 9749 6-3 Additional Notes: 22rv r r TT  = = = 6A.2 Angular Velocity  Definition: Angular velocity  is defined as the rate of change of angular displacement. d dt = θ Note: • Its unit is rad s−1. • It is a vector quantity. The direction pointing out of the page along the axis of rotation when the rotation is in the plane of the page anti-clockwise. • Its magnitude is called angular speed. Check Your Understanding 2 What is the angular speed of the second hand of a clock? A 0.105 rad s–1 B 9.55 rad s–1 C 0.0523 rad s–1 6A.3 Relationship between Angular Velocity  and Linear Velocity v From s r=θ and d dt = θ , we can show that:     = r s dt d and r is a constant = dt ds r 1 = r v i.e. rv = where v is the linear velocity or tangential velocity, r is the radius of the circular path,  is the angular velocity. v Or

ANDERSON SERANGOON JUNIOR COLLEGE PHYSICS 9749 6-4 Additional Notes: Note: • v has units m s−1 • It is a vector quantity. • Its magnitude is called linear speed or tangential speed. • Its direction is tangent to the circle and therefore always perpendicular to the radius. Check Your Understanding 3 What is the relationship between linear velocity v, radius of rotation r and angular velocity ω? 6A.4 Uniform Circular Motion • An object is undergoing uniform circular motion if it moves round a circle with constant speed. • The time it takes to go one revolution is called the period T. For 1 period T, the angular displacement  = 2 rad. π2 tT  == • The number of revolutions per unit time is called the frequency f, with unit Hertz (Hz). Since it takes time T to make one revolution, or Check Your Understanding 4 An object moving with uniform circular motion has a constant velocity. TRUE / FALSE   2 1 == Tf fω 2= =

ANDERSON SERANGOON JUNIOR COLLEGE PHYSICS 9749 6-5 Additional Notes: Example 1 Consider the second hand of a clock. Three points A, B and C lie on the second hand. (a) State and explain the point which has the largest (i) angular velocity , (ii) linear speed v. (i) angular velocity Since all points lie on the same radius as they move, they experience the same change in angular displacement per unit time. Hence they have the same angular velocity as t  =   (ii) linear speed o Since rv = and that the points have the same , the radius is directly proportional to linear speed. o Point C has the largest linear speed since it is the further away from the centre of circular motion. (b) Point C is 2.0 cm from the centre of the circle, calculate the (i) angular velocity of point C, [0.105 rad s−1] (ii) linear velocity of point C. [2.09  10−3 m s−1] A B C Key questions on objective Do the 3 points experience the same change in angular displacement over time? Yes. The 3 points lie along the same radius as they move. Hence their change in angular displacement per unit time is the same. Do the 3 points experience the same change in linear displacement over time? No. C experiences the largest change in linear displacement since it traces a larger circular arc as it moves. Conversely, A experiences the smallest change in displacement. (i) Key question to solve question What general knowledge do we know about the rotation of the second hand of a clock? It takes 60 s to go through one round, i.e. For t = T = 60 s,  = 2 Note that r does not refer to the radius of the revolving object. Rather, it refers to the radius of the circular path described by the point considered.

ANDERSON SERANGOON JUNIOR COLLEGE PHYSICS 9749 6-6 Additional Notes: (i) 22 0.10560tT  = = = = ππ rad s-1 (ii) rv = = 0.020 x 0.105 = 2.09  10−3 m s-1 Example 2 Singapore lies approximately on the Earth’s equator. The radius of the Earth is about 6370 km. Calculate the (i) angular velocity and [7.27  10−5 rad s−1] (ii) linear speed, of Singapore. [463 m s−1] (i) Angular velocity π2 T = = π2 24 60 60 = 51027.7 − rad s-1 (West to East) (ii) rv = = 356370 10 7 27 10. −   = 463 m s -1 (faster than speed of sound in air!) 6B Centripetal Acceleration and Centripetal Force For a uniform circular motion • The speed of the object is constant. • However, its direction is always changing. Hence its velocity is changing. • The object thus has an acceleration. By Newton’s second law, it must have a resultant force acted on it. (i) Key question to solve question What general knowledge do we know about the rotation of the Earth about its axis? It takes 24 hours for Earth to spin 1 round about its own axis, i.e. For t = T = 24 hours,  = 2 The speed of an object can be constant, but its velocity is changing, if the object’s motion changes direction. Recall that the SI unit of time is the second.

ANDERSON SERANGOON JUNIOR COLLEGE PHYSICS 9749 6-7 Additional Notes: vs vr vv   =  =  Alternatively, • By Newton’s first law, a body move