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## Chapter 8 Rotational Equilibrium and Dynamics

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**Torque(t)**A quantity that measures the ability of a force to rotate an object around some axis. t Greek letter ‘tau’**Net Torque produces rotation.**Just like a net force causes motion/acceleration.**Torque depends upon a force and the length of the lever arm.****How easily an object rotates depends not only on ‘how**much’ force is applied, but also where the force is applied.**Lever Arm**The perpendicular distance from the axis of rotation to a line drawn along the direction of the force.**Torque also depends upon the angle between a force and the**lever arm.**t = Fd(sinq)**Metric Units of torque are Newton-meters (N-m)**Torque, like displacement and force, is a vector quantity.**Since torque is a rotational motion it has two directions:**Counterclockwise (CCW), which is positive.**• Clockwise (CW), which is negative.**Ex 1: Sean has a flat and is changing his tire, to remove**the lug nuts he applies a force of 25 N on the tire iron at an angle of 87o. What is the torque produced if the tire iron is 0.6 m long?**G: d = 0.6 m, F= 25 N, q = 87o**U: t = ? E: t = Fd(sinq) S: t = (25 N)(0.6 m)(sin 87o) S: t =14.98 N-m**Ex 2: What angle produces a torque of 400 N-m, if the force**applied is 505 N at a distance from the axis of rotation of 0.82 m?**G: t = 400 N-m, F = 505 N, d =0.82 m**U: q = ?**G: t = 400 N-m, F = 505 N, d =0.82 m**U: q = ? E:**G: t = 400 N-m, F = 505 N, d =0.82 m**U: q = ? E:**Point mass**Where all the mass is assumed to be located in one point.**Center of Mass**The point at which all the mass of a body can be considered to be concentrated when analyzing translational motion.**Rotational and translational motion can be combined.**We use the the center of mass, as a reference, to analyze its translation motion.**Center of Gravity**The position at which the gravitational force acts on an extended object as if it were a point mass.**Toppling**• If the Center of Gravity (CG) is above the support area, then the object will remain upright. • If the CG extends outside the support area, the object will topple.**Unstable Equilibrium**Is when any movement/ displacement of a balanced object lowers the CG.**Stable Equilibrium**• Is when any motion/displacement of a balanced object raises its CG.**Neutral Equilibrium**• Is when any motion/displacement of a balanced object neither raises nor lowers its CG.**Moment of Inertia (MOI)**The tendency of a body to rotate freely about a fixed axis to resist a change in rotational motion.**MOI is the rotational analog of mass.**Very similar to mass, but MOI is not an intrinsic property of an object.**It depends upon the object’s mass and the distribution of**mass around the axis of rotation.**The farther the mass is, on average from the axis of**rotation, the greater the the object’s MOI and the more difficult it is to rotate the object.**Calculating the MOI**Pg 285: Table 8-1 has equations/formulas for a few common shapes. M – mass in kilograms R – radius in meters l – length in meters**For an object to be in complete equilibrium, requires zero**net force and zero net torque.**If the net force is zero the object is in translational**equilibrium. (1st Condition of Equilibrium)**If the net torque is zero, than it is in rotational**equilibrium. (2nd Condition for Equilibrium)**Newton’s 2nd Law for Rotation**F = ma We know in a rotational system, torque is a function of force, also:**1. MOI replaces the mass.**2. Acceleration is replaced with angular acceleration.**Remember that if the net torque is zero, that the object can**still be rotating, just at a constant velocity.**Ex 4: Simon decides to ride the ‘Gravitron’, it has a**radius of 3 m, if his mass is 79.4 kg. What is the net torque produced when his angular acceleration is 7 rads/sec2?**G: R = 3 m, M= 79.4 kg, a = 7 rads/sec2**U: tnet = ? E: tnet = Ia We are going to assume he is a point mass.**MOI, I = MR2**S:tnet = MR2a S: tnet = (79.4)(3)2(7) S: tnet = 5002.2 N-m**Ex 5: In, 1995, a fully functional pencil with a mass of 24**kg and a length of 2.74 m was made. Suppose this pencil is suspended at its midpoint and a force of 1.8 N is applied perpendicular to its end, causing it to rotate. What is the angular acceleration of the pencil?