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NOTES 13 - Topic 2 - Mechanics

NOTES 13 - Topic 2 - Mechanics - -------------------------------------------------------------------------------------------- 2 .4.1 Draw a vector diagram to show that the acceleration of a particle moving with uniform speed in a circular path is directed toward the center of the circle;

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NOTES 13 - Topic 2 - Mechanics

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  1. NOTES 13 - Topic 2 - Mechanics --------------------------------------------------------------------------------------------- 2.4.1 Draw a vector diagram to show that the acceleration of a particle moving with uniform speed in a circular path is directed toward the center of the circle; An object in Uniform Circular Motion (UCM): ...is moving in a circular path with constant speed; ...is moving with a constantly changing velocity; ...has a constant, inward-directed acceleration caused by a constant, inward-directed force; vt1 radius radius vt2

  2. 2.4.2 Apply the expression for centripetal acceleration. Kinematics of UCM Centripetal Acceleration (ac)... perpendicular to velocity; directed toward center of circle; ac = vT2 r -1 = 4 π2 r T-2 Radial Acceleration (ie., along the radius) ...another name for ac; Frequency (of revolution): f = (# revs)/time eg., f = revs s-1 = s-1= Hertz (Hz); Period (of revolution): T = time per revolution; T = s rev -1; f = 1/T; frequency is the inverse of period, and vice versa; UCM speed = distance / time [distance = circumference = 2 π r] ...so... vc = 2 π r T -1 = 2 π f

  3. 2.4.3 Identify the force producing circular motion in various situations. Dynamics of UCM Newton’s 2nd Law for Centripetal Force: ΣFc = mac = mvc2 r -1 = 4 Π 2 r m T-2 “Inertial Effects” “Inertial Separator” (aka “centrifuge”) - device which rotates and causes objects to move according to their mass (inertia); Example 1 - water is separated from clothes during the spin cycle of modern washing machines; Example 2 - U-235 (less plentiful bomb isotope) is separated from U-238 (more plentiful non-explosive isotope); (There is no such force as centrifugal force...it is a fictitious force, a forbidden word, and is on the “hit list”; it was invented to explain inertial effects.) “Artificial Gravity” can be caused by rotating a spacecraft in freefall; at the correct rate of rotation, the inertia of the bodies in the spacecraft and centripetal force exerted by the walls of the spacecraft simulate the feeling of gravity and let the inhabitants “walk around normally”.

  4. 2.6.4 Solve problems for objects moving in circular paths with uniform speed; Sample Problem 1 - horizontal circle (show solution in NB) Estimate the force a person must exert on a 0.30 kg ball on a cord to make it revolve in horizontal circle of radius 1.5 m at 2.0 Hz. Given: m = 0.30 kg; r = 1.5 m; f = 2.0 Hz; T = Unknown: F = ? Equation:

  5. Sample Problem 2 - vertical circle, constant speed (show solution in NB) A ball of mass 0.300 kg is swung by a cord in a vertical circle of 1.44 m diameter at a constant speed of 4.00 ms-1. (A) What is the tension in the cord at the top of its path? A. Given: m = 0.300 kg; r = 0.72 m; v = 4.00 ms-1; Unknown: FTA = ? Equation:

  6. (B) What is the tension in the cord at the bottom of the arc? Given:m = 0.300 kg; v = 4.00 ms-1; r = 0.72 m; Unknown: FTB = ? Equation:

  7. (C) At what speed would the object have to travel at the top so that the object is just in freefall? Given:m = 0.300; r = 0.72 m; Unknown: v = ? Equations:

  8. Sample Problem 3 - “artificial gravity” (show solution in NB) A “space habitat” is a cylinder that is 10. km long and 2.0 km in diameter. At what speed must it rotate to maintain an “artificial gravity” inertial effect of 9.8 ms-2? Give answer in ms-1 and rpm. Given: r = 1.0 km = 1.0 x 103 m; g = 9.8 ms-2; Unknown: v = ? Equation:

  9. Sample Problem 4 - “roller coaster” (show solution in NB) At what minimum speed must a roller coaster be moving when upside down in a loop to keep the people from falling out? Assume the radius of curvature is 10.0 m. Given: r = 10.0 m Unknown: v = ? Equation:

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