Senior Mathematics C - Dynamics Circular Motion

Circular Motion 12 Dynamics (notional time 30 hours) - Focus The approach used throughout this topic should bring together concepts from both vectors and calculus. Students should use a vector and/or a calculus approach to develop an understanding of the motion of objects that are subjected to forces.

Circular Motion 13 Subject matter derivatives and integrals of vectors (SLEs 1, 2, 3, 13) Newton’s laws of motion in vector form applied to objects of constant mass (SLEs 2–15) application of the above to: (SLEs 4–12, 14, 15) straight line motion in a horizontal plane with variable force vertical motion under gravity with and without air resistance projectile motion without air resistance simple harmonic motion (derivation of the solutions to differential equations is not required) circular motion with uniform angular velocity.

Circular Motion 14 Suggested learning experiences The following suggested learning experiences may be developed as individual student work, or may be part of small-group or whole-class activities. 1.Given the position vector of a point as a function of time such as r (t) = t i + t2 j + sin t k determine the velocity and acceleration vectors. 2.Given the displacement vector of an object as a function of time, by the processes of differentiation, find the force which gives this motion. 3.Given the force on an object as a function of time and suitable prescribed conditions, such as velocity and displacement at certain times, use integration to find the position vector of the object. 4.Investigate the motion of falling objects such as in situations in which: resistance is proportional to the velocity, by considering the differential equation resistance is proportional to the square of velocity, by considering the differential equation where k is a positive constant. 5.Model vertical motion under gravity alone; investigate the effects of the inclusion of drag on the motion. 6.Develop the equations of motion under Hooke’s law; verify the solutions for displacement by substitution and differentiation; relate the solutions to simple harmonic motion and circular motion with uniform angular velocity. 7.From a table of vehicle stopping distances from various speeds, calculate (a) the reaction time of the driver and (b) the deceleration of the vehicle, which were assumed in the calculation of the table. 8.Model the path of a projectile without air resistance, using the vector form of the equations of motion starting with a = -g j where upwards is positive. 9.  Use the parametric facility of a graphing calculator to model the flight of a projectile. 10.Investigate the flow of water from a hose held at varying angles, and model the path of the water. 11.Investigate the motion of a simple pendulum with varying amplitudes. 12.Investigate the angle of lean required by a motorcycle rider to negotiate a corner at various speeds. 13.Use the chain rule to show that the acceleration can be written as if the velocity, v, of a particle moving in a straight line is given as a function of the distance, x. 14.Investigate the speed required for a projectile launched vertically to escape from the earth’s gravitational field, ignoring air resistance but including the variation of gravitational attraction with distance. 15.Use detectors or sensors to investigate problems, e.g. rolling a ball down a plank. 16.  Use spreadsheets to investigate problems.

Circular Motion 15 Suggested learning experiences The following suggested learning experiences may be developed as individual student work, or may be part of small-group or whole-class activities. 1.Given the position vector of a point as a function of time, determine the velocity and acceleration vectors. 2.Given the displacement vector of an object as a function of time, by the processes of differentiation, find the force which gives this motion. 12.Investigate the angle of lean required by a motorcycle rider to negotiate a corner at various speeds.

Circular Motion 16 How Circular Motion was taught in Please view video clip at:

Circular Motion 17 Mr. Stephen Kulik, a teacher at the Orange County High School of the Arts, introduces his Block class to the idea of circular motion. Please view video clip at:

Circular Motion 18 The Preferred Approach

Circular Motion 19 Position Vector r r = (r cos  ) i + (r sin  ) j For uniform circular motion, d  / dt =  and r = (r cos  t) i + (r sin  t) j

Circular Motion 110 Velocity Vector v r = (r cos  t) i + (r sin  t) j v = r = (-r  sin  t) i + (r  cos  t) j Note that v. r = 0, and that v is perpendicular to r.. v

Circular Motion 111 Acceleration a is given by A. B. C. D. E. All of the above.

Circular Motion 112 Acceleration Vector a r = (r cos  t) i + (r sin  t) j v = r = (-r  sin  t) i + (r  cos  t) j a = v = r = (-r  2 cos  t) i + (-r  2 sin  t) j. a...

Circular Motion 113 Acceleration Vector a r = (r cos  t) i + (r sin  t) j v = r = (-r  sin  t) i + (r  cos  t) j a = v = r = (-r  2 cos  t) i + (-r  2 sin  t) j. a... | v | = v = r   = v / r | a | = a = r  2 = v 2 /r

Circular Motion 114 Centripetal Force F c F c = ma = (-mr  2 cos  t) i + (-mr  2 sin  t) j FcFc a = r  2 = v 2 /r F c = ma = mr  2 = mv 2 /r

Circular Motion 115 Student Exercise Design the banking of a road around a curve with radius 250 metres in an area where the speed limit is 100 km/hr.

Circular Motion 116 Design the banking of a road around a curve with radius 250 metres in an area where the speed limit is 100 km/hr. n = (n sin  ) i + (n cos  j W = -mg j j i

Circular Motion 117 Design the banking of a road around a curve with radius 250 metres in an area where the speed limit is 100 km/hr. F net = (n sin  ) i + (n cos  j - mg j j i

Circular Motion 118 Design the banking of a road around a curve with radius 250 metres in an area where the speed limit is 100 km/hr. F net = (n sin  ) i + (n cos  j - mg j F c = (mv 2 /r) i F net = F c j i

Circular Motion 119 Design the banking of a road around a curve with radius 250 metres in an area where the speed limit is 100 km/hr. F net = (n sin  ) i + (n cos  j - mg j F c = (mv 2 /r) i F net = F c n cos  – mg = 0 -> n cos  = mg n sin  = mv 2 /r j i

Circular Motion 120 Design the banking of a road around a curve with radius 250 metres in an area where the speed limit is 100 km/hr. n cos  = mg n sin  = mv 2 /r j i n sin  = mv 2 n cos  rmg

Circular Motion 121 Design the banking of a road around a curve with radius 250 metres in an area where the speed limit is 100 km/hr. j i tan  = v2v2 rg

Circular Motion 122 Design the banking of a road around a curve with radius 250 metres in an area where the speed limit is 100 km/hr. j i tan  = (100 x 1000 / 3600 ) x 9.8 tan  = …  = 17.5 o

Circular Motion 123 Coming Attractions Please view video clip at:

Circular Motion 124 Circular Motion 2 1. A 2 kg ball on a string is rotated about a circle of radius 10 m. The maximum tension allowed in the string is 50 N. What is the maximum speed of the ball? 2. During the course of a turn, an automobile doubles its speed. How much additional frictional force must the tyres provide if the car safely makes around the curve? 3. A satellite is said to be in geosynchronous orbit if it rotates around the earth once every day. For the earth, all satellites in geosynchronous orbit must rotate at a distance of 4.23 x 10 7 metres from the earth's centre. What is the magnitude of the acceleration felt by a geosynchronous satellite? 4. The maximum lift provided by a 500 kg aeroplane is N. If the plane travels at 100 m/s, what is its shortest possible turning radius? 5. A popular daredevil trick is to complete a vertical loop on a motorcycle. This trick is dangerous, however, because if the motorcycle does not travel with enough speed, the rider falls off the track before reaching the top of the loop. What is the minimum speed necessary for a rider to successfully go around a vertical loop of 10 metres?

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