Rotational Motion and the Law of Gravity Lecture Notes Physics 2053 Rotational Motion and the Law of Gravity

Rotational Motion and the Law of Gravity Topics 7-04 Centripetal Acceleration 7-05 Newtonian Gravitation 7-06 Kepler’s Laws Rotational Motion and the Law of Gravity

Centripetal Acceleration Uniform circular motion: motion in a circle of constant radius at constant speed Instantaneous velocity is always tangent to circle. v2 v1 Rotational Motion and the Law of Gravity

Centripetal Acceleration Radial Acceleration: Similar Triangles v2 -v1 v2 Dt Dq r2 Dr Dq Divide by Time r1 Dv v1 Divide by time Uniform Circular Motion Simular Triangles Centripetal Acceleration Rotational Motion and the Law of Gravity

Centripetal Acceleration In uniform circular motion the acceleration is called the centripetal, or radial, acceleration. It is perpendicular to the velocity and points towards the center of the circle. v ar r Rotational Motion and the Law of Gravity

Rotational Motion and The Law of Gravity Is it possible for an object moving with a constant speed to accelerate? Explain. A) Yes, although the speed is constant, the direction of the velocity can be changing. B) No, if the speed is constant then the acceleration is equal to zero. C) No, an object can accelerate only if there is a net force acting on it. D) Yes, if an object is moving it can experience acceleration Rotational Motion and the Law of Gravity

Centripetal Acceleration Problem A jet plane travelling 525 m/s pulls out of a dive by moving in an arc of radius 6.00 km. What is the plane’s acceleration? Rotational Motion and the Law of Gravity

Rotational Motion and The Law of Gravity An object moves in a circular path at a constant speed. Compare the direction of the object's velocity and acceleration vectors. A) The vectors are perpendicular. B) Both vectors point in the same direction. C) The vectors point in opposite directions. D) The question is meaningless, since the acceleration is zero. Rotational Motion and the Law of Gravity

Rotational Motion and The Law of Gravity What type of acceleration does an object moving with constant speed in a circular path experience? A) free fall B) constant acceleration C) linear acceleration D) centripetal acceleration Rotational Motion and the Law of Gravity

Centripetal Acceleration For an object to be in uniform circular motion, there must be a net force acting on it. The radial force on the ball is provided by the string v ar Fr There is no centrifugal force acting on the ball This radial force is called a centripetal force Rotational Motion and the Law of Gravity

Centripetal Acceleration The speed of an object in Uniform Circular Motion m T r M Mg Rotational Motion and the Law of Gravity

Centripetal Acceleration Problem A 0.45 kg ball, attached to the end of a horizontal cord, is rotated in a circle of radius 1.3 m on a frictionless horizontal surface. If the cord will break when the tension in it exceeds 75 N, what is the maximum speed the ball can have? Rotational Motion and the Law of Gravity

Centripetal Acceleration Motion in a vertical circle T mg The tension in the string when the ball is at the top. r Rotational Motion and the Law of Gravity

Centripetal Acceleration Motion in a vertical circle The tension in the string when the ball is at the bottom. r T mg Rotational Motion and the Law of Gravity

Centripetal Acceleration Problem A bucket of mass 2.00 kg is whirled in a vertical circle of radius 1.10 m. At the lowest point of its motion the tension in the rope supporting the bucket is 25.0 N. (a) Find the speed of the bucket. Rotational Motion and the Law of Gravity

Centripetal Acceleration Problem (con’t) A bucket of mass 2.00 kg is whirled in a vertical circle of radius 1.10 m. (b) How fast must the bucket move at the top of the circle so that the rope does not go slack? Rotational Motion and the Law of Gravity

Rotational Motion and The Law of Gravity A pilot executes a vertical dive then follows a semi-circular arc until it is going straight up. Just as the plane is at its lowest point, the force on him is A) less than mg, and pointing up. B) less than mg, and pointing down. C) more than mg, and pointing up. D) more than mg, and pointing down. Rotational Motion and the Law of Gravity

Centripetal Acceleration Maximum Speed in horizontal turn N r m ax fmax mg Rotational Motion and the Law of Gravity

Rotational Motion and The Law of Gravity A car goes around a curve of radius r at a constant speed v. What is the direction of the net force on the car? A) toward the curve's center B) away from the curve's center C) toward the front of the car D) toward the back of the car Rotational Motion and the Law of Gravity

Rotational Motion and The Law of Gravity A car goes around a curve of radius r at a constant speed v. Then it goes around a curve of radius 2r at speed 2v. What is the centripetal acceleration of the car as it goes around the second curve, compared to the first? A) four times as big B) twice as big C) one-half as big D) one-fourth as big Rotational Motion and the Law of Gravity

Centripetal Acceleration Problem How large must the coefficient of static friction be between the tires and the road if a car is to round a level curve of radius 85 m at a speed of 95 km/h? Rotational Motion and the Law of Gravity

Centripetal Acceleration q Friction on a banked road v a q f mg q r Rotational Motion and the Law of Gravity

Centripetal Acceleration Friction on a banked road N v q When f a When f q mg When No Friction Rotational Motion and the Law of Gravity

Centripetal Acceleration v Turning a banked curve with no friction q a q mg Rotational Motion and the Law of Gravity

Rotational Motion and The Law of Gravity The banking angle in a turn on the Olympic bobsled track is not constant, but increases upward from the horizontal. Coming around a turn, the bobsled team will intentionally "climb the wall," then go lower coming out of the turn. Why do they do this? A) to give the team better control, because they are able to see ahead of the turn B) to prevent the bobsled from turning over C) to take the turn at a faster speed D) to reduce the g-force on them Rotational Motion and the Law of Gravity

Centripetal Acceleration Weight Reading on scale is the normal force N mg Scale Rotational Motion and the Law of Gravity

Centripetal Acceleration Apparent Weight at the Earth’s Surface At the North Pole: NN mg At the Equator: v NE mg Rotational Motion and the Law of Gravity

Centripetal Acceleration A space station is in the shape of a hollow ring 450 m in diameter. Gravity is simulated by rotating the ring. Find the speed in revolutions per minute needed in order to simulate the Earth’s gravity. R v N Rotational Motion and the Law of Gravity

Centripetal Acceleration The speed in revolutions per minute v N R = 225 m Rotational Motion and the Law of Gravity

Newtonian Gravitation Gravitational Force: Gravitational Force is the mutual force of attraction between any two objects in the Universe. m F R M Universal Gravitational Constant Rotational Motion and the Law of Gravity

Newtonian Gravitation Gravitational Potential Energy associated with an object of mass m at a distance r from the center of the Earth is r m ME Rotational Motion and the Law of Gravity

Newtonian Gravitation Escape velocity An object projected upward from the Earth’s surface with a large enough speed will soar off into space and never return. This speed is called the Earth’s escape velocity. Rotational Motion and the Law of Gravity

Newtonian Gravitation Problem Calculate the acceleration due to gravity on the Moon. The Moon’s radius is 1.74 x 106 m and its mass is 7.35 x 1022 kg. Rotational Motion and the Law of Gravity

Newtonian Gravitation Problem A hypothetical planet has a mass 1.66 times that of Earth, but the same radius. What is g near its surface? Rotational Motion and the Law of Gravity

Newtonian Gravitation Speed of a Satellite v m F R M Rotational Motion and the Law of Gravity

Rotational Motion and The Law of Gravity Two planets have the same surface gravity, but planet B has twice the radius of planet A. If planet A has mass m, what is the mass of planet B? Rotational Motion and the Law of Gravity

Newtonian Gravitation Problem A certain neutron star has five times the mass of our Sun packed into a sphere about 10 km in radius. Estimate the surface gravity on this monster. Rotational Motion and the Law of Gravity

Newtonian Gravitation The satellite is kept in orbit by its speed – it is continually falling, but the Earth curves from underneath it. Without gravity With gravity Earth Rotational Motion and the Law of Gravity

Rotational Motion and The Law of Gravity Compared to its mass on the Earth, the mass of an object on the Moon is A) the same. B) less. C) more. D) half as much. Rotational Motion and the Law of Gravity

Newtonian Gravitation Problem Calculate the force of Earth’s gravity on a spacecraft 12,800 km (2 Earth radii) above the Earth’s surface if its mass is 1350 kg. Rotational Motion and the Law of Gravity

1. All planets move in elliptical orbits with the Kepler’s Laws Kepler’s Three Laws: 1. All planets move in elliptical orbits with the Sun at one of the focal points. Rotational Motion and the Law of Gravity

2. A line drawn from the Sun to any planet sweeps Kepler’s Laws Kepler’s Three Laws: 2. A line drawn from the Sun to any planet sweeps out equal areas in equal time intervals. Area 1 Area 2 Area 1 = Area 2 Rotational Motion and the Law of Gravity

3. The square of the orbital period of any planet is Kepler’s Laws Kepler’s Three Laws: 3. The square of the orbital period of any planet is proportional to the cube of the average distance from the planet to the Sun. T R Rotational Motion and the Law of Gravity

Kepler’s Laws T r Rotational Motion and the Law of Gravity

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