3 Lab 3 Rotation Rotational Motion of Rigid

實驗 3：轉動-剛體的轉動運動 Lab. 3 : Rotation -Rotational Motion of Rigid Body 實驗目的：測量作用在圓盤上之力矩所產生的角加速 度和角速度的大小 l證明剛體繞固定軸所做的轉動運動，亦遵守牛頓第二 運動定律。 l並探討轉動物體的能量守恆關係。 Object: To measure angular acceleration and angular velocity of a rotating rigid disk applied by a torque . l. To prove the rotational motion of a rigid body obey the Newtonian 2 nd motional law. Physics. NTHU l. The rotational motion also obeys the conservation of. MFTai-戴明鳳 energy.

Hyper. Physics Most fundamental concepts are subtracted from the web site: http: //hyperphysics. phy-astr. gsu. edu/hbase/hframe. html Physics. NTHU MFTai-戴明鳳

Rotational Dynamics http: //hyperphysics. phy-astr. gsu. edu/hbase/hframe. html Physics. NTHU MFTai-戴明鳳

Circular / Rotational Motion For circular motion at a constant speed v, the centripetal acceleration of the motion can be derived. Physics. NTHU http: //hyperphysics. phy-astr. gsu. edu/hbase/circ. html#rotcon MFTai-戴明鳳

Basic Rotational Quantities l The angular displancement is defined by: l For a circular path it follows that the angular velocity is l The angular acceleration: i. e. , Tangential acceleration: l Centripetal acceleration: Physics. NTHU MFTai-戴明鳳

Radian and Conversions l is a pure number, but commonly is given the artificial unit, radian. l One radian: is the angle subtended by an arc length equal to the radius of the arc (s = r). l Conversion between degrees and radians Physics. NTHU MFTai-戴明鳳

Angular Displacement l The angular displacement is defined as the angle the object rotates through during some time interval. l This is the angle that the reference line of length r sweeps out. Fig 10. 2 Physics. NTHU MFTai-戴明鳳

Average Angular Speed & Instantaneous Angular Speed l The average angular speed, , of a rotating rigid object is the ratio of the angular displacement to the time interval. l The instantaneous angular speed is defined as the limit of the average speed as the time interval approaches zero. Units: radians/sec, rad/s or s-1 (radians have no dimensions) l > 0: for increasing (counterclockwise) l < 0: for decreasing (clockwise) Physics. NTHU l MFTai-戴明鳳

Average and Instantaneous Angular Acceleration l The average angular acceleration, , of an object is defined as the ratio of the change in the angular speed to the time it takes for the object to undergo the change. l The instantaneous angular acceleration is defined as the limit of the average angular acceleration as the time goes to 0 l Units of angular acceleration are rad/s 2 or s-2 since radians have no dimensions. Physics. NTHU MFTai-戴明鳳

Angular Motion, General Notes l When a rigid object rotates about a fixed axis in a given time interval, every portion on the object rotates through the same angle in a given time interval and has the same angular speed and the same angular acceleration. – So , , all characterize the motion of the entire rigid object as well as the individual particles in the object. Directions of l Strictly speaking, the speed and acceleration ( , ) are the magnitudes of the velocity and acceleration vectors. l The directions are actually given by the right-hand rule. Physics. NTHU Fig 10. 3 MFTai-戴明鳳

Rotational Kinematics l Under constant angular acceleration, the motion of the rigid object can be described using a set of kinematic equations. – These are similar to the kinematic equations for linear motion. – The rotational equations have the same mathematical form as the linear equations. Rotational Kinematic Equations Physics. NTHU MFTai-戴明鳳

Comparison Between Rotational and Linear Equations Physics. NTHU MFTai-戴明鳳

Relationship Between Angular and Linear Quantities l Displacements l Speeds l Accelerations l Every point on the rotating object has the same angular motion. l Every point on the rotating object does not have the same linear motion. Physics. NTHU MFTai-戴明鳳

Speed Comparison l The linear velocity is always tangent to the circular path. – called the tangential velocity l The magnitude is defined by the tangential speed. Fig 10. 4 Physics. NTHU MFTai-戴明鳳

Tangential Acceleration Centripetal Acceleration l Fig 10. 5 An rotating rigid object traveling in a circle, even though it moves with a constant speed, will have an centripetal acceleration: Physics. NTHU MFTai-戴明鳳

Resultant Acceleration l Tangential component of the acceleration change speed l Centripetal component of the acceleration change direction l Total acceleration: Physics. NTHU MFTai-戴明鳳

Notes for Speed and Acceleration l All points on the rigid object will have the same angular speed, but not the same tangential speed. l All points on the rigid object will have the same angular acceleration, but not the same tangential acceleration. l The tangential quantities depend on r, and r is not the same for all points on the object. Physics. NTHU MFTai-戴明鳳

Torque vs. Force = I vs. F = ma l Forces can cause a change in linear motion. – Described by the Newton’s Second Law l Torque motion. can cause a change in rotational – The effectiveness of this change depends on the force and the moment arm. – The change in rotational motion depends on the torque. – Also obey the Newton’s 2 nd Motional Law Physics. NTHU MFTai-戴明鳳

Torque l Be the vector product or cross product of two vectors r and F. l SI units of torque: N. m l Torque is a force multiplied by a distance, it is very different from work and energy (unit: Joules). Fig 10. 12 Physics. NTHU MFTai-戴明鳳

Definition of Vector Product l Any two vectors, l The vector product and = Fig 10. 13 l The direction of C is given by the right-hand rule. Physics. NTHU MFTai-戴明鳳

Conditions for Equilibrium l An object at equilibrium has no net influences to cause it to move, either in translation (linear motion) or rotation. The basic conditions for equilibrium are: Physics. NTHU MFTai-戴明鳳

Rotational-Linear Parallels Physics. NTHU MFTai-戴明鳳

The Rigid Object In Equilibrium Fig 10. 15 (a) & (b) Physics. NTHU MFTai-戴明鳳

Moment of Inertia, Rotational inertia l Moment of inertia (rotational inertia) ~ mass for linear motion. l It appears in the relationships for the dynamics of rotational motion. l The moment of inertia must be specified with respect to a chosen axis of rotation. l For a point mass the moment of inertia is just the mass times the square of perpendicular distance to the rotation axis, I = mr 2. l That point mass relationship becomes the basis for all other moments of inertia since any object can be built up from a collection of point masses. Physics. NTHU MFTai-戴明鳳

Moment of Inertia l Dimensions: ML 2, SI units: kg. m 2 l It is more easily to calculate the moment of inertia of an object by assuming it is divided into many small volume elements, each of mass Dmi. l For continuous body, If r is constant, the integral can be evaluated with known geometry, otherwise its variation with Physics. NTHU position must be known. MFTai-戴明鳳

Physics. NTHU MFTai-戴明鳳

Physics. NTHU MFTai-戴明鳳

Fig 10. 7 Physics. NTHU MFTai-戴明鳳

Parallel Axis Theorem The I (rotational inertia) of any object about an axis through its center of master (C. M. ), Icm, is the minimum moment of inertia for an axis in that direction in space. l The moment of inertia about any axis parallel to that axis through the C. M. is given by the parallel axis theorem. l l The expression added to the Icm will be recognized as the moment of inertia of a point mass - the moment of inertia about a parallel axis is the center of mass moment plus the moment of inertia of the entire object treated as a point mass at Physics. NTHU the center of mass. MFTai-戴明鳳

Analogy between Rotational & Linear Kinetic Energies, KErotational & KELinear For a given fixed axis of rotation, the rotational kinetic energy can be expressed in the form a rotating object is analogous to KELinear and can be expressed in terms of the moment of inertia and angular velocity. l The total kinetic energy of an extended object can be expressed as the sum of Physics. NTHU the translational kinetic energy of the center of mass and the rotational kinetic MFTai-戴明鳳 energy about the center of mass. l KErotational

Work-Energy Principle l The expressions for KErotational & KELinear kinetic energy can be developed in a parallel manner from the work-energy principle. l Consider the following parallel between a constant torque exerted on a flywheel with moment of inertia I and a constant force exerted on a mass m, both starting from rest. Physics. NTHU MFTai-戴明鳳

Rotational Kinetic Energy of the rigid object l l The total rotational kinetic energy is the sum of the energies of all its particles: [units: Joules (J)]: where I is called the moment of inertia l Be similar to the linear kinetic energies associated with motion (K = 1/2 mv 2) l Be not a new type of energy, the form is different because it Physics. NTHU is applied to a rotating object. MFTai-戴明鳳

For the linear case, starting from rest, the acceleration from Newton's second law is equal to the final velocity divided by the time and the average velocity is half the final velocity, showing that the work done on the block gives it a kinetic energy equal to the work done. l For the rotational case, also starting from rest, the rotational work is τ θ and l the angular acceleration α given to the flywheel is obtained from Newton's second law for rotation. l The angular acceleration is equal to the final angular velocity divided by the time and the average angular velocity is equal to half the final angular velocity. l It follows that the rotational kinetic energy given to the flywheel is equal to the work done by the torque. Physics. NTHU l MFTai-戴明鳳

Work-Energy Principle This is a general principle which can be applied specifically to rotating objects. l For pure rotation, the net work is equal to the change in rotational kinetic energy and for a constant torque, the work can be expressed as l For a net torque, Newton's 2 nd law for rotation gives Combining this expression gives a useful relationship for describing rotational motion. Physics. NTHU MFTai-戴明鳳

Rolling Objects without Slipping The K. E. (kinetic energy) is divided between linear kinetic energy and rotational kinetic energy. l Another key is that for rolling without slipping, the linear velocity of the center of mass is equal to the angular velocity times the radius. l Physics. NTHU MFTai-戴明鳳

Rotation Vector Examples This is an active graphic. Click on any example. http: //hyperphysics. phy-astr. gsu. edu/hbase/hframe. html Physics. NTHU MFTai-戴明鳳

原理(Principle) 物體對旋轉軸轉動(Body rotates about a fixed rotational axis) : 角位移(angular displacement): [rad] [360 o = 2 radians] 角速度(angular velocity): w = dq/dt [rad/s] 角加速度(angular acceleration): = dw/dt = d 2 /dt 2 [rad/s 2] 外力矩(torque) = r x F (F: 作用力, r : 外力與原點距離) [m. N] then or = I = d. L/dt 比率常數: 對轉動軸之轉動慣量(rotational inertia): I [kg. m 2] I = Simiri 2 = rr 2 d. V mi: 質量, ri: 質量與原點距離, r: 均勻質量密度, r: 與原點距離, d. V: 體積單元 力矩(torque) = constant, = constant 等角加速度運動 力矩 = 0, L = constant, w = constant 等角速度運動 Physics. NTHU MFTai-戴明鳳

1. 小輪軸(small rotational axis) 2. 次小輪軸(other rotational axis) 3. 外側插栓(outer insert pin) 4. 內側插栓(Inner insert pin) 5. 送風口(Wind port) 6. 滑輪(frictionless pulley) T = mg 圖 2 氣墊桌裝置 (Fig. 2 Air Bearing Turntable) 7. 輪軸架(Pulley support) 8. 圓盤(Turntable Disc) 9. 砝碼(weights) 10. 摩擦橡皮圈(Resist elastic band) Physics. NTHU 11. 氣墊桌(Air Table) MFTai-戴明鳳

頂視圖 (Top view) 1. 小輪軸(small rotational axis) 2. 次小輪軸(other rotational axis) 3. 外側插栓(outer insert pin) 4. 內側插栓(Inner insert pin) 7. 輪軸架(Support of rotation axis ) 8. 圓盤(Disc) 9. 砝碼(weights) mg – T = ma T = mg – ma a= r T = mg - m r Tension, T = mg T is the force applied by the weight , mg. Physics. NTHU MFTai-戴明鳳

mg – T = ma T = mg – ma a= r T = mg - m r Physics. NTHU MFTai-戴明鳳