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UNIFORM CIRCULAR MOTION In Uniform Circular Motion, an object moves with constant speed in a circular path. v,t = a,c = a,c = v,t 2 / r r = When an object completes one lap ( or ), it covers a distance of =. - Time for one cycle ( ) in [ ] - Inverse of Period ( ) in [ ] v,t = = Period is seconds/cycle, frequency is cycles/second. RPM: Revs per Minute: f = RPM / 60 EXAMPLE 1: Calculate the period, frequency, and speed of an object moving in uniform circular motion (radius 10 m) if: (a) it completes 100 cycles in 60 seconds; (b) it takes 3 minute to complete 1 cycle. EXAMPLE 2: The car below takes 10 s to go from A to B, at constant speed. If the semi-circle has radius of 5 m, find its: (a) period; (b) tangential velocity; (c) centripetal acceleration. A B NOTE: Even though the object s speed is constant, its direction changes, therefore its velocity changes and. Constant speed, but NON-ZERO centripetal acceleration (tangential velocity changes direction). Page 2
MORE: UNIFORM CIRCULAR MOTION PRACTICE 1: A Ping-Pong ball goes in a horizontal circle (radius 5 cm) inside a red cup twice per second. Find its: (a) period; (b) speed; (c) centripetal acceleration. U. CIRCULAR MOTION a, C = v, T 2 / r v, T = 2 π r / T = 2 π r f f = 1 / T = RPM / 60 EXAMPLE 1: One way to simulate gravity (or create artificial gravity ) in a space station is to spin it. If a cylindrical space station (diameter = 500 m) is spun about its central axis, at how many revolutions per minute (rpm) must it turn so that the outermost points have acceleration equal to the acceleration due to gravity at the surface of the Earth? PRACTICE 2: A 3kg rock spins horizontally at the end of a 2-m string at 90 rpm. Calculate its: (a) speed; (b) acceleration. Page 3
CENTRIPETAL FORCES In linear motion, we have forces in the X & Y axes. Now, we ll have forces in the axis. - Before, we had ΣFX = max and ΣFY = may. Now, we have (remember a,c = v,t 2 / r) - When writing ΣFC, forces towards the center are, forces away are, and tangential forces don t get listed: EXAMPLE 1: A small 3 kg object on top of a frictionless table is attached to the end of a 2 m string, as shown. If the object spins once every 2 seconds, calculate the tension on the string. PRACTICE 1: For the situation above, suppose the string breaks if its tension exceeds 50 N. Calculate the maximum speed that the object can attain without breaking the string. U. CIRCULAR MOTION a, C = v, T 2 / r v, T = 2 π r / T = 2 π r f f = 1 / T = RPM / 60 EXAMPLE 2: Some crazy fighter pilot (70 kg) in some movie does a nose dive that is nearly circular (radius 300 m). If his speed at the bottom of the dive is 80 m/s, find his: (a) centripetal acceleration; (b) apparent weight. NOTE: Centripetal Force is not a force in nature, but simply an indication that a force acts in the centripetal direction. Page 4
CENTRIPETAL FORCES: VERTICAL EXAMPLE: A Ferris Wheel of radius 50m takes 30 seconds to make a full cycle. An 80 kg guy rides on it. Calculate his: (a) speed and centripetal acceleration; (b) apparent weight at the bottom; (c)* apparent weight at the top. EXAMPLE: What is the minimum speed that a rollercoaster cart can have at the top of a vertical loop of radius 10 m so that the passengers won t fall, even at the absence of restraints (eg. seat belts)? Similar question: Find v,min for a bucket in a vertical loop so that water doesn t fall while the bucket is at the top. PRACTICE: A pendulum is made from a light, 2 m-long rope and a 5-kg small object. When you release the object from rest as of a certain height, it swings from side to side, attaining a maximum speed of 10 m/s. At the object s lowest point: (a) Draw a Free Body Diagram. (b) Find the magnitude of its acceleration. (c) Find the tension on the rope. Page 5
CENTRIPETAL FORCES: FLAT & BANKED CURVES EXAMPLE 1: Find the maximum speed that a 800 kg car can have while going around a flat curve of radius 50 m (without slipping) if the coefficient of friction between the car and the road is 0.5. EXAMPLE 2: (a) Find the maximum speed that a 800 kg car can have while going around a banked, frictionless curve of radius 50 m that makes an angle of 37 o with the horizontal. What would happen if the car moves: (b) slower; (c) faster? PRACTICE 1: You are designing a highway curve to allow cars to turn, without any banking, at a maximum speed of 50 m/s. The average coefficient of friction between cars and asphalt, for dry roads, is roughly 0.7. What radius would this curve have to have, for this to be possible? PRACTICE 2: For the radius you just found, how much would you have to bank the same curve, in order to attain the same maximum speed, but at the absence of friction? Page 6
THE UNIVERSAL LAW OF GRAVITATION The force of gravity is not just the Earth pulling down on objects. Because of Action/Reaction, it is a mutual attraction. - Newton s Universal Law of Gravitation states that : - Attraction Force w = FG = - Univ. Grav. Constant G = m 3 kg -1 s -2 - Forces are along between objects. - Distance r is between. EXAMPLE 1: Two identical solid spheres of mass 10 kg and 60 cm in diameter are placed side by side without any space between them. Calculate their attraction force. PRACTICE 1: The International Space Station (ISS) has mass 4.5 x 10 5 kg and is 370 km above the Earth. The Earth has mass 5.98 x 10 24 kg and radius 6370 km. Find the ISS s weight. EXAMPLE 2: (a) Three objects are fixed in place as shown. Calculate the NET gravitational force on M in terms of G, M, L. (b) Suppose 2M comes lose: In which direction would it accelerate? Would this resulting acceleration be constant? 2M M 3M x = 0 x = 3L x = 5L NOTE: Notice how Gravitational forces are very weak for non-planetary stuff. So we ll ignore them unless otherwise stated. Page 7
MORE: GRAVITATIONAL FORCE FG = G m1 m2 / r 2 G = 6.67x10-11 m 3 kg -1 s -2 PRACTICE 1: (a) How hard does the Earth (5.98 x 10 24 kg) pull on the Moon (7.35 x 10 22 kg) if they are 3.85 x 10 8 m apart? (b) How hard does the Sun (1.99 x 10 30 kg) pull on the Moon if they are 1.50 x 10 11 m apart? PRACTICE 2: Calculate the net force acting on the Moon when it is aligned with the Sun and the Earth, as shown below. Use the values given and forces found in EXAMPLE 1 (above). EXTRA: What acceleration (magnitude and direction) does the Moon have as a result of this net force? E M S EXAMPLE 1: Three objects are lined up as shown. Objects of mass 2M and 3M are 10L apart. How far from 2M, in terms of L, would M have to be in order to experience no net force? 3M M 2M Page 8
GRAVITATION: NET FORCE IN 2D EXAMPLE 1: Four small 100-kg spheres are arranged as shown, forming a square of sides 10 cm. Calculate the magnitude and direction (angle) of the net force acting on the sphere on the bottom left corner. Page 9
PRACTICE 1: Three small 100-kg spheres are arranged as shown, forming a triangle of equal 10-cm sides. Calculate the magnitude and direction of the net force acting on the bottom sphere. Page 10
LAW OF GRAVITATION: ACCELERATION DUE TO GRAVITY Remember g refers to the acceleration due to free fall at a planet s surface (or asteroid, etc.). FG = G m1 m2 / r 2 G = 6.67x10-11 m 3 kg -1 s -2 - We can use the Universal Law of Gravitation to derive an expression for a planet s g based on its properties: g FAR from surface: ( r = ) g,far = g NEAR the surface: ( h r ) g,near = EXAMPLE 1: The Earth has mass 5.98 x 10 24 kg and radius 6.37 x 10 6 m. Calculate the acceleration due to free fall at the top of Mount Everest, 8,500 m above water. PRACTICE 1: The Moon has a diameter of 3,475 km, and the acceleration due to free fall on its surface is roughly one sixth that of the Earth. Calculate the mass of the Moon. PRO-TIP: If using the g,near equation to find R or h, find r first. Then use r = to find R or h. EXAMPLE 2: The International Space Station experiences a free fall acceleration of 8.77 m/s 2 as it rotates around the Earth. Calculate its height above the Earth s surface. Page 11
SATELLITE MOTION: CIRCULAR ORBITS A Satellite is any object that orbits another. Examples: (1) Moon around the Earth, (2) Earth around the Sun. - For a Satellite launched from the Earth, its orbit (shape of path) depends on its : v = 0 V CIRCULAR V ESCAPE Object s speed Projectile Motion Elliptical Orbit Escape Circular Orbit Circular orbits are simpler. Planets have orbits, but often we simplify ( nearly circular ). - In Circular orbits, the orbital speed and height of a satellite are related by: GRAV. CONSTANTS VSAT = G = 6.67 x 10-11 M E = 5.97 x 10 24 kg R E = 6.37 x 10 6 m - Combining VSAT and v = 2 π r / T (speed in circular motion): r = TSAT = (orbital period, circular) For every height a satellite may have, there is an exact corresponding speed it must have to maintain that height. - Orbital speed, period, and height are interdependent. As height increases, v, T. - Notice how a satellite s motion does NOT depend on its own mass (m). EXAMPLE: The International Space Station (ISS) is in nearly circular orbit at 370 km above the Earth. Calculate its orbital: (a) speed; (b) period. What would happen if it was moving: (c) slightly slower/faster; (d) much faster; (e) much slower? NOTE: Astronauts in the I.S.S. are NOT weightless; they are constantly falling towards the Earth. In fact, g,iss = ~8.7 m/s 2. Page 12
MORE: CIRCULAR ORBITS ProTip #1: When solving for the speed to put an object in orbit, find vsat (circular). PRACTICE 1: (a) How fast would you have to throw an object, horizontally from the ground, for it to become a low-orbit satellite around the Earth? (b) What orbital period (in hours) would it have? GRAV. CONSTANTS G = 6.67 x 10-11 M E = 5.97 x 10 24 kg R E = 6.37 x 10 6 m CIRCULAR ORBITS v, SAT = v, T = 2 π r / T T, SAT = 2 π r / v, SAT T, SAT = 2 π r 3/2 / r = R + h ProTip #2: When solving for R or h, first find r, then use r = R + h to find R or h. EXAMPLE 1: How high above the Earth s surface must a satellite moving at 5,000 m/s be in order to have circular orbit? PRACTICE 2: A satellite in circular orbit takes 30 hours to go around the Earth. Calculate its height above the Earth. When a satellite stays in place relative to the Earth, its orbit is called and. - The Earth s Period around itself is ( ); around the Sun is ( ). EXAMPLE 2: How high above the Earth must you place a satellite so it is constantly flying directly over the same spot? Page 13