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Transcript
Gravitation and Rotational Motion Hannah Packer Things to know: Kepler’s Laws Newton’s Law of Universal Gravitation Gravitational force Gravitational constant Gravitational Field Angular Displacement Angular Velocity Angular Acceleration Lever Arm Torque Center of Mass Centrifugal Force Key Concepts for Gravitation: 1. Kepler’s 1st Law- Path of the planets are ellipses, with the Sun at one focus. 2. Kepler’s 2nd Law- An imaginary line from the sun to a planet sweeps out equal areas in equal time areas. 3. Kepler’s 3rd LawT is equivalent to the period of a planet revolving around the Sun, and r is the average distance from the sun. Newton’s Law of Universal Gravitation: this says that objects attract other objects with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. Other important equations include: Period of a Planet Orbiting the Sun – this is equal to 2 x pi times the square root of the orbital radius cubed, divided by the product of the universal gravitational constant and the mass of the sun. Speed of a Satellite Orbiting Earth- this is equal to the square root of the universal gravitational constant times the mass of Earth, divided by the radius of the orbit. Period of a Satellite Orbiting Earth- this is equal to 2 x pi times the square root of the radius of the orbit cubed, divided by the product of the universal gravitational constant and the mass of Earth. Gravitational Field- this is equal to the universal gravitational constant times the object’s mass, divided by the square of the distance from the object’s center. Key Concepts for Rotational Motion: As an object rotates, the change in the angle is called angular displacement. Counterclockwise rotation is designated as positive while clockwise motion is negative. The angular velocity of an object is angular displacement divided by the time taken to make the displacement. Angular acceleration is defined as the change in angular velocity divided by the time required to make the change. For a given applied force, the change in angular velocity depends on the lever arm, which is the perpendicular distance from the axis of rotation to the point where the force is exerted. Torque- is a measure of how effectively a force causes rotation. The magnitude of torque is the product of the force and the lever arm. Torque is measured in Newton-meters. p.s. (multiply by sin theta if used at an angle) Newton’s Second Law for Rotational Motion: states that angular acceleration is directly proportional to the net torque and inversely proportional to the moment of inertia. Center of Mass- this is the point on an object that moves in the same way that a point particle would move. Centrifugal Force- This is the apparent force that seems to pull on a moving object, but does not exert a physical outward push on it, and is observed only in rotating frames of reference. Multiple Choice If the radius of the Earth were increased by a factor of 2 and its mass remained the same, then the acceleration due to gravity on the Earth would…? A) Increase by a factor of 4 B) Reduce by a factor of 4 C) Reduce by a factor of 2 D) Increase by a factor of 2 Free Response Figure 1 shows an object with several forces acting on it. The pivot point is at O. F1 = 10 N, and is at a distance of 0.25 m from O, where = 80 F2 = 7.0 N, acting perpendicular to the object, at a distance of 1.25 m from O F3 = 12 N, is 0.60 m from O, and acts at = 40 from the horizontal Find the total (net) torque on the object. Answers: Multiple choice= Reduce by a factor of 4 Free Response= -6.6 N m