Rotational Motion PDF

SH1685 Rotational Motion 𝑑 𝜃=...

SH1685 Rotational Motion 𝑑 𝜃= 𝑟 I. Introduction Their involved factor is the radius (𝑟), which All concepts in Physics are applicable to one another, from 2. Angular Displacement (∆𝜽). It is the measurement where speed, position, and so on. In this topic, we shall be translating the angle through which a point, line, or body has rotated in linear motion to rotation. a specified direction, about a specified axis. It may also refer 1. Angular Position (𝜽). Before, our Position is denoted by 𝑑, to the difference between the initial (𝜃0 ) and final (𝜃𝑓 ) and this is usually associated with linear motion. But, it is angular positions. It is also similar to the linear also applicable to circles, and even rotational motion. Here, displacement, ∆𝑥. It has the same relationship with it, using it denotes the orientation of a rotating object in free space. the same equation from angular position, It also represents the direction of the displacement vector ∆𝑥 ∆𝜃 = drawn from the origin (or axis) to the location of the 𝑟 revolving object. Unit is in radians (rads), which is 3. Angular Velocity (𝝎). Just like the linear velocity, angular equivalent to fractional values of 𝜋. To convert values from velocity also shows the rate of change in an object’s position. degrees to radians, What differs from the linear velocity is that it measures the 𝜋 𝑟𝑎𝑑𝑖𝑎𝑛 = 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 × change in object’s angular displacement per unit of time 180° ∆𝜃 And converting radians to degrees, 𝜔= 180° 𝑡 𝑑𝑒𝑔𝑟𝑒𝑒 = ( ) × 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 The formula to show the relationship of both linear and 𝜋 angular velocities also uses the radius as its factor, Rotation and revolution are both key concepts in this topic. 𝑣 They both mean alike in some ways, but different as well. 𝜔= 𝑟 Rotation is the spinning movement of an object without actually leaving its position, kind of like how a top spins as Since 𝜔 is a vector, its value is dependent on the direction it it is being unwind by the rope, or how our planet has night takes. Angular velocity is different from uniform circular and day. Revolution, on the other hand, is the movement of motion and tangential speed, however. Uniform circular an object around the axis. This means that object is said to motion has the formula, 𝐶 be revolving if it is moving around a point, or axis, while an 𝑣= object is said to be rotating if it spins in its place. 𝑡 Where: Relationship-wise, the relationship between distance and 𝐶 = 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑜𝑓 𝑐𝑖𝑟𝑐𝑙𝑒 (2𝜋𝑟) angular position is, 𝑡 = 𝑝𝑒𝑟𝑖𝑜𝑑 (𝑡𝑖𝑚𝑒) 08 Handout 1 *Property of STI Page 1 of 7  SH1685 It shows how fast an object moves in a uniform manner In Physics, inertia is a state of complete equilibrium. Mass is a (constant speed and angular velocity) while taking a curved factor of inertia, meaning, the heavier the object is, the more path. Tangential speed, as the name implies, is the speed of force is required to start changing its current state. Moreover, an object perpendicular to its radius, mass is innate inertia, while momentum is moving inertia. 𝑣𝑡 = 𝜔𝑟 In rotational motion, it is called moment of inertia (sometimes ∆𝜃𝑟 called rotational inertia), where it is dependent on both mass 𝑣𝑡 = and the position of the object’s axis. The axis can exist anywhere 𝑡 from within the object itself or outside its system. The moment 4. Angular Acceleration (𝜶). This is also similar with the of inertia changes because of the changes in position its axis is rectilinear acceleration, only this time, it involves placed at. By definition, rotational inertia is the resistance of a angular velocity. rotating body to change its current state of balance, which in this It shows the changes in an object’s angular velocity per case, is its rotation. unit of time, ∆𝜔 𝛼= 𝑡 It also uses the radius as its proportionality factor, 𝑎 𝛼= 𝑟 And by relating it to angular velocity, 𝜔2 𝛼= 𝑟 And, with uniform circular motion, Source: https://scripts.mit.edu/ 4𝜋 2 𝑟 𝛼= 2 Mathematically, 𝑡 𝑖 = 𝑚𝑟 2 Also, 𝛼 is a vector. Its value is dependent on the direction Where: it takes. In this case, if 𝜔 is going in a different direction 𝑖 = 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 than 𝛼, then 𝜔 is negative. Likewise, if 𝜔 follows 𝛼, then 𝑚 = 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑜𝑏𝑗𝑒𝑐𝑡 𝜔 is positive. 𝑟 = 𝑟𝑎𝑑𝑖𝑢𝑠 II. Rotational Equilibrium Moment of Inertia 08 Handout 1 *Property of STI Page 2 of 7  SH1685 But there are instances where an object contains more than one Solid Cylinder (1) point in its radius. These points represent mass, but each Axis is at cylinder’s point has a different amount of mass. So, to put it simply, center 𝑖 = ∑𝑚𝑖 𝑟𝑖2 𝑚𝑟 2 Where: 𝑖= 2 𝑖 = 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 𝑚𝑖 = 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑜𝑏𝑗𝑒𝑐𝑡 Axis is at height’s center 𝑟𝑖 = 𝑟𝑎𝑑𝑖𝑢𝑠 𝑚𝑟 2 𝑚ℎ2 𝑖= + Because every object has a different axis location, computation 4 12 for the rotational inertia differs. Below is the table for the moment of inertia for each common figure. Thin Rod ROTATIONAL FIGURE INERTIA Axis is at the end, Hoop perpendicular to the rod Axis is in hollow center 𝑖 = 𝑚𝑟 2 𝑚𝑙 2 𝑖= 3 Axis is in diameter of hoop 𝑖 = (𝑚𝑟 2 )/2 Axis is at center of the rod, perpendicular to its Annular Cylinder length 𝑚 2 𝑚𝑙 2 𝑖= (𝑟 + 𝑟22 ) 𝑖= 2 1 12 08 Handout 1 *Property of STI Page 3 of 7  SH1685 Sphere 𝑎=0 Rotational motion also has static equilibrium, which will be covered by the next topics. Spherical shell III. Rotational Dynamics 2𝑚𝑙 2 Torque 𝑖= Imagine that moment when you pushed the door too hard, it 3 swung open very rapidly. You just saw the door swung past you very fast, then hit the wall with a very loud, “THUD!” That door has just experienced torque. To engine and car aficionados, it is the performance of a cycled car engine. To them, a car with the best acceleration shift is the Solid Sphere car with the highest torque capacity. That is because torque is the force acting on a rotating body, which is done at a certain 2𝑚𝑙 2 𝑖= distance away from its point of origin. Another phrase used to 5 describe torque is “moment of force”. Torque uses two (2) standard formulas, 𝜏 = 𝐹𝑟 𝑠𝑖𝑛𝜃 Plane Slab Where: 𝐹 = 𝑓𝑜𝑟𝑐𝑒 𝑚(𝑎2 + 𝑏 2 ) 𝑟 = 𝑙𝑒𝑣𝑒𝑟 𝑎𝑟𝑚 𝑖= 12 𝜃 = 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 Image source: https://en.wikipedia.org/wiki/List_of_moments_of_inertia 𝜏 = 𝑖𝛼 Where: Static Equilibrium Remembering Free Body Diagram, it is a graphical 𝑖 = 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 representation of all forces acting on the object. If all forces are 𝛼 = 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 equal, then the system is said to have attained static equilibrium Torque is the counterpart of linear force. To achieve because the system is not moving. equilibrium during rotational motion, Mathematically speaking, to achieve static equilibrium, 𝜏𝑛𝑒𝑡 = ∑𝜏𝑖 ∑𝐹 = 0 𝑤ℎ𝑒𝑟𝑒 𝜏𝑛𝑒𝑡 = 0 08 Handout 1 *Property of STI Page 4 of 7  SH1685 Therefore, if an object achieves rotational equilibrium, it is Rotational Translational Notes either not rotating, or it is rotating at a constant angular velocity. 𝜃 = 𝜔𝑎𝑣𝑒 𝑡 𝑥 = 𝑣𝑎𝑣𝑒 𝑡 Rotational Quantities and Kinematics 𝜔𝑓 = 𝜔0 + 𝛼𝑡 𝑣𝑓 = 𝑣0 + 𝑎𝑡 Observe the table in the opposite page.The values in the given 1 1 𝜃 = 𝜔0 + 𝛼𝑡 2 𝑥 = 𝑣0 + 𝑎𝑡 2 both 𝑎 & 𝛼 are table are the same equations from the previous topics. But now, 2 2 constant these are translated to rotational motion. Some of these variables 2 2 may have different units, forms, or both, while others still retain 𝜔𝑓2 = 𝜔𝑓2 + 2𝛼𝜃 𝑣𝑓 = 𝑣𝑓 + 2𝑎x their units and forms. Acceleration is the only variable that remained the same. 𝑆 = 𝜃𝑟 Angular Work, Kinetic Energy, and Momentum If that is the case, by rewriting the formula, we also get the Rotational motion also performs work and has kinetic energy angular displacement, and momentum. To perform angular work (𝑊𝜏 ), which is work 𝑆 done by a rotating system, 𝜃= 𝑟 𝑊𝜏 = 𝑖𝛼𝜃 Where: The net torque is equivalent to net force multiplied by the radius that force is being translated to, 𝑖 = 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 𝜏𝑛𝑒𝑡 = 𝐹𝑛𝑒𝑡 𝑟 𝛼 = 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 Combining it with our work formula, along with arc length Recalling that 𝑖𝛼 = 𝜏, derivation, we can derive a new formula, 𝑊𝜏 = 𝜏𝑛𝑒𝑡 𝜃 𝑆 Where: 𝑊𝜏 = (𝐹𝑛𝑒𝑡 𝑟) ( ) 𝑟 𝜏𝑛𝑒𝑡 = 𝑛𝑒𝑡 𝑡𝑜𝑟𝑞𝑢𝑒 ∆𝜃 = 𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑊𝜏 = 𝐹𝑛𝑒𝑡 𝑆 Angular work also follows the same pattern as linear work. Now, recall the formula for determining the final angular velocity using the initial velocity, angular position, and angular acceleration, 𝑊 = 𝐹𝑑, in which force is being translated over a certain distance. 𝜔𝑓2 = 𝜔02 + 2𝛼𝜃 Angular work, however, translates work within a certain angular Solving for 𝛼𝜃, displacement in which it is known as the arc length (usually 2𝛼𝜃 = 𝜔𝑓2 − 𝜔02 stylized as the symbol 𝑆). Arc length is defined by the formula, 08 Handout 1 *Property of STI Page 5 of 7  SH1685 Whenever we push a round table, ride the bike, or even just open 𝜔𝑓2 − 𝜔02 𝛼𝜃 = the door, we all exert torque. If the torque we exert is stronger 2 than the other torques opposing the torque we exert, then the Recalling the angular work formula, and by deriving 𝛼𝜃, object rotates, such as the strong slam of the door if we open it 𝑊𝜏 = 𝑖𝛼𝜃 with a great heave. So, the greater the net torque exerted within 𝑊𝜏 a short time, the stronger the angular momentum, which is 𝛼𝜃 = shown mathematically, 𝑖 ∆𝐿 If we are to apply it to the derived formula for 𝛼𝜃, 𝜏𝑛𝑒𝑡 = 𝑊𝜏 𝜔𝑓2 − 𝜔02 ∆𝑡 = If we are to rearrange this equation as such, 𝑖 2 2 𝜏𝑛𝑒𝑡 ∆𝑡 = ∆𝐿 𝑖𝜔𝑓 − 𝑖𝜔02 𝑊𝜏 = We get the exact same formula as we would write down the 2 impulse – momentum theorem, The equation that we have derived above is the work –energy 𝐹∆𝑡 = ∆𝑝 theorem for rotational motion. As you may recall, kinetic energy arises from the changes in the net work done by the system. By By analyzing it, this fundamental equation is the rotational applying analogy to linear kinetic energy, angular kinetic energy version of Newton’s Second Law. And, we can take this to a (𝑇𝜏 ) is, higher level. If we can assume that the net torque is equal to zero 1 (0), as shown below, 𝑇𝜏 = 𝑖𝜔2 𝜏𝑛𝑒𝑡 = 0 2 Now, knowing that the moment of inertia (𝑖) has the same and, considering the torque-angular momentum equation, ∆𝐿 implication as mass (𝑚), if we are to simply multiply it with 𝜏𝑛𝑒𝑡 = angular velocity (𝜔), we get angular momentum (𝐿), ∆𝑡 𝐿 = 𝑖𝜔 So, if we substitute the common values, ∆𝐿 This equation is analogous to the linear momentum equation, =0 𝑝 = 𝑚𝑣 ∆𝑡 We can see that it is also familiar with another Law, the Law of The only difference is their respective units. Linear momentum Conservation of Linear Momentum. This time, this is for uses kg-m/s2, whereas angular momentum uses kg-m2/s. This Angular Momentum. If we see that the change in angular implies that larger masses have bigger momenta than smaller momentum (∆𝐿) is zero, bodies. Likewise, having a faster angular velocity also lends its hand in increasing angular momentum. ∆𝐿 = 0 08 Handout 1 *Property of STI Page 6 of 7  SH1685 Then, Hewitt, Paul G. (2007). Conceptual physics (3rd ed.). California, Addison- 𝐿𝑓 = 𝐿0 Wesley Publishing Company Khan Academy. (2017). Centripetal force and acceleration intuition. If both momenta have equal magnitude, then angular momentum Retrieved 2017, April 4 from Khan Academy: is constant all throughout the system (𝐿 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡). https://www.khanacademy.org/science/physics/centripetal-force- and-gravitation/centripetal-acceleration-tutoria/v/centripetal-force- References: and-acceleration-intuition Bauer, W., & Westfall, G. D. (2016). General physics 1 (2nd ed.). Nave, C. R. (2016). Circular motion. Retrieved from The Georgia State Columbus, OH: McGraw-Hill Education. University’s HyperPhysics: Bauer, W., & Westfall, G. D. (2016). General physics 1 (2nd ed.). 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Quezon City: Phoenix Publishing house, Inc. Elert, G. (2017). Impulse and momentum. Retrieved 2017, March 31 from the Physics Hypertextbook: http://physics.info/momentum/summary.shtml Freedman, R. A., Ford, A. L., & Young, H. D. (2011). Sears and zemansky's university physics (with Modern physics) (13th ed.). Addison-Wesley. Giambattista, A., Richardson, B. M., & Richardson, R. C. (2007). College physics (2nd Ed.). New York: The McGraw-Hill Companies, Inc. Halliday, D., Resnick, R. & Walker, J. (2007). Fundamentals of physics (5th ed.). New York: John Wiley and Sons, Inc. 08 Handout 1 *Property of STI Page 7 of 7