Holt Physics Chapter 7: Rotational Motion and the Law of Gravity I. Section 7-1: Measuring Rotational Motion A. When something spins it undergoes “rotational motion”. When something spins around a single point it is called “circular motion”. B. We measure how fast something spins not in m/s (different points on the object are spinning at different velocities) but by measuring the angle described in a given time period. C. Angles can be measured in radians (rad) 1. The radian is the ratio of the arc length (s) to the radius (r) of a circle (insert fig. 7-1 here) 2.
The radian is a “pure number” with no units (the abbreviation “rad” is always used) 3. Conversions: 360o = 2? rad 360o = 6. 28 rad ?(rad) = ? /180o ? (deg) ?(rad) = . 0174533 ? (deg) (insert fig. 7-3) D. Angular displacement describes how much an object has rotated relative to a reference line (insert fig 7-4) Angular Displacement ?? = ? s/r angular displacement = change in arc length/radius E. Watch your sign! ? is considered positive when rotating COUNTERclockwise (when viewed from above). Therefore an angle of ?? rad = -1?? rad F. Angular Speed (? “omega”) describes the rate of rotation. Average angular speed is measured in radians per second. Angular Speed ?avg = ?? /? t average angular speed = angular displacement/time G. Angular Acceleration (? = “alpha”) occurs when angular speed changes. Remember acceleration? a = velocity/time ?? Angular Acceleration ?avg = ? 2 – ? 1/t2 – t1 = ?? /? t average angular acceleration = change in speed/time H. “All points on a rotating rigid object have the same angular acceleration and angular speed. ” P. 250 II. Section 7-2: Tangential and Centripetal Acceleration A.
Relationships between angular and linear quantities 1. Objects in circular motion have a “tangential speed” 2. This is not the “tangent” from trigonometry but the “tangent” that intersects a circle at exactly one point tangent circle 3. “tangential speed” is an object’s speed along a line tangent to its circular path 4. Objects further from the center of the circle have a higher tangential speed (these quantities are directly related, i. e. as one gets larger so does the other) 5.
Think of horses flying off a carousel on tangential paths Tangential Speed vt = r? tangential speed = radius x angular speed 6. Tangential Acceleration is tangent to the circular path Tangential Acceleration at = r? tangential acceleration = radius x angular acceleration B. Centripetal Acceleration 1. A change in an object’s direction must be described as an acceleration a. since a = v/t and velocity is a vector with magnitude and direction… b. by changing an object’s direction we change its velocity and, hence its acceleration! 2.
An object moving at a constant speed around a circular path has an acceleration due to its constantly changing direction. An acceleration of this type is called “centripetal acceleration” Centripetal Acceleration ac = vt2/r centripetal acceleration = (tangential speed)2/ radius ac = r? 2 centripetal acceleration = radius x (angular speed)2 3. Tangential and centripetal accelerations are perpendicular because centripetal acceleration always points towards the center of the circle 4. This fact enables us to “Find the total acceleration using the Pythagorean theorem” (p. 59) Insert fig. 7-9 III. Section 7-3: Causes of Circular Motion A. “Force That Maintains Circular Motion” 1. Whatever force is preventing an object from traveling a straight line (gravity or a tether perhaps) can be measured in Newtons and is called the “force that maintains circular motion” Force That Maintains Circular Motion Fc = mvt2/r Force = mass x (tangential speed2/radius) Fc = mr? 2 Force = mass x radius x angular speed2 B. Newton’s Law of Universal Gravitation states that gravitational force depends on the distance between two masses
Newton’s Law of Universal Gravitation Fg = (G)(m1m2/r2) gravitational force = gravitational constant x (product of masses/radius2) C. The gravitational constant (G) is defined as 6. 673 x 10-11 Newton meters squared per kilogram squared and has been proven experimentally D. Newton’s Law of Universal Gravitation is often referred to as the “inverse square law” as force decreases when distance increases (we say these two are inversely related) E. Gravitational force is localized at the center of a spherical mass