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Rotational Motion Kinematics of Rotation • As an object revolves around a fixed point, each part of the object (even though it follows a different arc) sweeps through the same angle ϴ. Angular Displacement • An object revolving about a fixed point will sweep through the same angle ϴ at every point of the object. • Each part of the object moves through an arc length (l) at some radial distance from the center (r). ϴ=l r • Both l and r are distances, so the angle has no unit. • When l = r, ϴ = 1 • This is 1 radian or rad To move btwn rad and degrees: ϴ (rad) = ϴ (degrees) 2π (rad) 360 (degrees) Degrees Radians in terms of π Radians in decimals 0 0 0 15 π/12 0.262 30 π/6 0.524 45 π/4 0.785 57.3 1.00 1.00 60 π/3 1.047 90 π/2 1.571 120 2π/3 2.094 180 π 3.142 270 3π/2 4.172 360 2π 6.283 Example 1 The Andromeda galaxy is a giant spiral cluster of stars whose mass is that of 300 thousand million Suns. You can see it with the naked eye as a faint elongated cloud in the night sky. Inasmuch as it subtends an angle of 4.1o , and is larger than our own galaxy at 163 x 103 light years in diameter., how far away is it in light years? Angular Speed ω = ϴ/t Units: rad/s, but also rpm (rotations per minute) and rev/s (revolutions per second) Relationship btwn angular and linear speeds v = rω ω must be in rad/s to use this eqn • Notice that angular speed of the object will be the same for an object, but the linear speed depends on how far away from the center of rotation the point on an object is. Example 2 The music on a compact disc (CD) is encoded on tiny raised ridges lying on a spiraling path (up to 5.4 km long) that winds outward from the center of the disc. A laser tracks along the path at a constant linear speed of 1.2 m/s and the fluctuating light reflected off the ridges carries the info lifted from the CD. Typically, the track begins 2.3 cm from the center and ends about 5.9 cm from the center. To keep the linear speed of the laser read-out unit constant, the angular speed at which the disc turns is continuously varied. Determine the angular speed of the disc at the beginning and end of the data path. Give your answers in both rad/s and rev/s. Angular Acceleration • Variations in ω are as common as variations in v. • Average changes in angular speed is called angular acceleration. αav = ∆ω/∆t Relationship between angular and linear accel • Linear (aka tangential accel) α = aT/ r All these accelerations! • ar arises from a change in the direction of the motion of an object (not a change in speed). • aT arises when both the direction and the linear/tangential speed changes. • α arises from a change in angular speed. True for a runner who speeds up around a turn or a monkey slowing its swing on a vine. Example 3 At a particular moment in a race, a car roaring around a turn with a radius of 50 m had an angular speed of 0.60 rad/s and an angular acceleration of 0.20 rad/s2. Compute its linear speed, centripetal acceleration, tangential acceleration and total linear acceleration at that moment. Stop to Think A ball on the end of a string swings in a horizontal circle once every second. State whether the magnitude of each of the following quantities is zero, constant (but not zero), or changing. a. b. c. d. e. Velocity Angular velocity Centripetal acceleration Angular acceleration Tangential acceleration Eqns of Constant Angular Acceleration ω = ωo + αt 2 ω = ωo + 2αϴ 2 ϴ = ωot + ½ 2 αt Example 4 Mounted in a bus is a 2.0 m diameter flywheel, a massive disk often used to store rotational energy. If it is accelerated from rest at a constant rate of 2.0 rpm per second, what will be the angular speed of a point on the rim of the flywheel after 5.0 s? Through what angle will that point have rotated? Rotational Equilibrium • Aristotle analyzed the practical problem of balances, levers and seesaws. • When a pivoted beam is balanced horizontally so it doesn’t revolve, it’s in rotational equilibrium. • Law of the Lever: unequal forces on a pivoted bar balance each other if F1r1 = F2r2 Example 5 A boy (mass 30 kg) wishes to play on a centrally pivoted seesaw with his dog Irving (mass 10 kg). When the dog sits 3.0 m from the pivot, where must the boy sit if the 6.5 m long board is to be balanced horizontally? Torque • The ability of a force to cause a rotation depends on 3 factors: 1. The magnitude of the force 2. The distance (r) from the pivot to the point at which the force is applied 3. The angle at which the force is applied • These three factors are incorporated into a single quantity called torque. • Torque from Latin torquere– “to twist” • Torque is the rotational equivalent of force • Radial line: a line from the pivot point (or fulcrum) to the point at which the force is applied. It extends beyond that point, too. • The angle, phi, is the angle btwn the radial line and the force vector. • Only the perpendicular component of the force has an effect on the object’s rotation. • Think about F2 and the doorway. τ = rF┴ • Units: Newton-meters (Nm) Alternate way • Line of action: the line extending to infinity on either side of the force vector. • Moment arm: the perpendicular line from the line of action to the pivot point. τ = r┴F They’re the same • τ = rF┴ = r (Fsinϕ) or • τ = r┴F = (rsinϕ)F τ = rFsinϕ • All 3 eqn’s work equally well. First two are usually more practical in problem solving, but the last one is how it’s usually expressed. • Be careful with the third, though, you’re not always using sine, depending on the angle they’ve given you. • Angle phi is generally given in degrees. You can keep it that way as long as your calculator is in DEGREES. If you convert it to radians, switch your calculator to radians. • Calculator settings rad vs. degrees only matters when using sin/cos/tan • Torque differs from force in a VERY important way. • Torque is measured/calculated about a particular point. Expressing torque without specifying from where is meaningless. • It can be calculated from any point– which would change radius and angle (phi). • In practice, we usually calculate torques from a hinge, pivot, or fulcrum. Example 6 Ryan is trying to open a stuck door. He pushes it at a point 0.75 m away from the hinges with a 240 N force directed 20o away from being perpendicular to the door. Pivot point is the hinges. What torque does Ryan exert? How could he exert more? What about dir’n? • CCW is + • CW is – • You must decide the sign by observing the dir’n torque works. Concept check It is hard to going if you try to start your bike with the pedal at the highest point. Why is this? Let’s say you’re riding up a steep hill standing on the pedals. At what point in the pedals’ motions do you get the most forward motion? Example 7 Lou uses a 20 cm long wrench to tighten a nut, turning it clockwise. The wrench handle is tilted 300 above the horizontal and Lou pulls straight down on the end with a force of 100 N. How much torque does Lou exert on the nut?