Physics for Scientists & Engineers, with Modern Physics, 4 th edition Giancoli

Piri Reis University 2011-2012/ Physics -I Physics for Scientists & Engineers, with Modern Physics, 4th edition Giancoli

Piri Reis University 2011-2012 Fall Semester Physics -I Chapter 10 RotationalMotion

Piri Reis University 2011-2012 LectureX I. Angular Quantities II. Constant Angular Acceleration III. Rolling Motion (Without Slipping) IV. Torque V. Rotational Dynamics; Torque and Rotational Inertia VI. Solving Problems in Rotational Dynamics VII. Rotational Kinetic Energy VIII. Angular Momentum and Its Conservation IX. Vector Nature of Angular Quantities

I. Angular Quantities In purely rotational motion, all points on the object move in circles around the axis of rotation (“O”). The radius of the circle is r. All points on a straight line drawn through the axis move through the same angle in the same time. The angle θ in radians is defined: where l is the arc length.

I. Angular Quantities Angular displacement: The average angular velocity is defined as the total angular displacement divided by time: The instantaneous angular velocity:

I. Angular Quantities The angular acceleration is the rate at which the angular velocity changes with time: The instantaneous acceleration:

I. Angular Quantities Every point on a rotating body has an angular velocityω and a linear velocity v. They are related:

I. Angular Quantities Therefore, objects farther from the axis of rotation will move faster.

I. Angular Quantities If the angular velocity of a rotating object changes, it has a tangential acceleration: Even if the angular velocity is constant, each point on the object has a centripetal acceleration:

I. Angular Quantities Here is the correspondence between linear and rotational quantities:

I. Angular Quantities The frequency is the number of complete revolutions per second: Frequencies are measured in hertz. The period is the time one revolution takes:

II. Constant Angular Acceleration The equations of motion for constant angular acceleration are the same as those for linear motion, with the substitution of the angular quantities for the linear ones.

III. Rolling Motion (Without Slipping) In (a), a wheel is rolling without slipping. The point P, touching the ground, is instantaneously at rest, and the center moves with velocity v. In (b) the same wheel is seen from a referenceframe where C is at rest. Now point P is moving with velocity –v. The linear speed of the wheel is related to its angular speed:

IV. Torque To make an object start rotating, a force is needed; the position and direction of the force matter as well. The perpendicular distance from the axis of rotation to the line along which the force acts is called the lever arm.

IV. Torque A longer lever arm is very helpful in rotating objects.

IV. Torque Here, the lever arm for FA is the distance from the knob to the hinge; the lever arm for FD is zero; and the lever arm for FC is as shown.

IV.Torque The torque is defined as:

V. Rotational Dynamics; Torque and Rotational Inertia Knowing that , we see that This is for a single point mass; what about an extended object? As the angular acceleration is the same for the whole object, we can write:

V. Rotational Dynamics; Torque and Rotational Inertia The quantity is called the rotationalinertia of an object. The distribution of mass matters here – these two objects have the same mass, but the one on the left has a greater rotational inertia, as so much of its mass is far from the axis of rotation.

V. Rotational Dynamics; Torque and Rotational Inertia The rotational inertia of an object depends not only on its massdistribution but also the location of the axis of rotation – compare (f) and (g), for example.

Rotational Dynamics: I m2 m1 m3 Rotational Motion

Rotational Dynamics: I d Rotational Motion

Parallel-axis Theorem d Rotational Motion

Parallel-axis Theorem Rotational Motion

VI. Solving Problems in Rotational Dynamics • Draw a diagram. • Decide what the system comprises. • Draw a free-body diagram for each object under consideration, including all the forces acting on it and where they act. • Find the axis of rotation; calculate the torques around it. • 5. Apply Newton’s second law for rotation. If the rotational inertia is not provided, you need to find it before proceeding with this step. • 6. Apply Newton’s second law for translation and other laws and principles as needed. • 7. Solve. • 8. Check your answer for units and correct order of magnitude.

VII. Rotational Kinetic Energy The kinetic energy of a rotating object is given by By substituting the rotational quantities, we find that the rotational kinetic energy can be written: A object that has both translational and rotational motion also has both translational and rotational kinetic energy:

VII. Rotational Kinetic Energy When using conservation of energy, both rotational and translational kinetic energy must be taken into account. All these objects have the same potential energy at the top, but the time it takes them to get down the incline depends on how much rotational inertia they have.

VII. Rotational Kinetic Energy The torque does work as it moves the wheel through an angle θ:

VIII. Angular Momentum and Its Conservation In analogy with linear momentum, we can define angular momentumL: We can then write the total torque as being the rate of change of angular momentum. If the net torque on an object is zero, the total angular momentum is constant. Therefore, systems that can change their rotational inertia through internal forces will also change their rate of rotation:

IX. Vector Nature of Angular Quantities The angular velocity vector points along the axis of rotation; its direction is found using a righthand rule:

IX. Vector Nature of Angular Quantities Angular acceleration and angular momentum vectors also point along the axis of rotation.

Summary of Chapter 10 • Angles are measured in radians; a whole circle is 2π radians. • Angular velocity is the rate of change of angular position. • Angular acceleration is the rate of change of angular velocity. • The angular velocity and acceleration can be related to the linear velocity and acceleration. • The frequency is the number of full revolutions per second; the period is the inverse of the frequency. • The equations for rotational motion with constant angular acceleration have the same form as those for linear motion with constant acceleration. • Torque is the product of force and lever arm. • The rotational inertia depends not only on the mass of an object but also on the way its mass is distributed around the axis of rotation. • The angular acceleration is proportional to the torque and inversely proportional to the rotational inertia. • An object that is rotating has rotational kinetic energy. If it is translating as well, the translational kinetic energy must be added to the rotational to find the total kinetic energy. • Angular momentum is • If the net torque on an object is zero, its angular momentum does not change.

Example 1 An automobile wheel has a radius of 42 cm. If a car drives 10 km, through what angle has the wheel rotated? a) In revolutions b) In radians c) In degrees

Solution of Ex.1 Note distance car moves = distance outside of wheel moves a) Find N: Known: s = 10 000 m, r = 0.42 m b) Find q in radians Known: N c) Find q in degrees Known: N = 3 789 q = 2.38 x 104 rad. q = 1.36 x 106 deg

Example 2 A race car engine can turn at a maximum rate of 12 000 rpm. (revolutions per minute). a) What is the angular velocity in radians per second. b) If helipcopter blades were attached to the crankshaft while it turns with this angular velocity, what is the maximum radius of a blade such that the speed of the blade tips stays below the speed of sound. DATA: The speed of sound is 343 m/s

Solution of Ex.2 a) Convert rpm to radians per second = 1256 radians/s b) Known: v = 343 m/s, w = 1256 rad./s Find r = 0.27 m

Example 3. A pottery wheel is accelerated uniformly from rest to a rate of 10 rpm in 30 seconds. a) What was the angular acceleration? (in rad/s2) b)How many revolutions did the wheel undergo during that time?

Solution of Ex.3 First, find the final angular velocity in radians/s. a) Find angular acceleration = 0.0349 rad./s2 b) Find number of revolutions: Known wi=0, wf =1.047, and t = 30 First find Dq in radians = 2.5 rev.

Solution of Ex.3 b) Find number of revolutions: Known wi=0, wf =1.047, and t = 30, First find Dq in radians = 2.5 rev.

Example 4. A coin of radius 1.5 cm is initially rolling with a rotational speed of 3.0 radians per second, and comes to a rest after experiencing a slowing down of a = 0.05 rad/s2. a) Over what angle (in radians) did the coin rotate? b) What linear distance did the coin move?

Solution of Ex.4 a) Find Dq, Given wi= 3.0 rad/s, wf = 0, a = -0.05 rad/s2 = 90 radians = 90/2p revolutions b) Find s, the distance the coin rolled Given: r = 1.5 cm and Dq = 90 rad = 135 cm

Example 5. A space-station is constructed like a barbell with two 1000-kg compartments separated by 50 meters that spin in a circle (r=25 m). The compartments spins once every 10 seconds. • What is the acceleration at the extreme end of the compartment? Give answer in terms of “g”s. • If the two compartments are held together by a cable, what is the tension in the cable?

Solution of Ex.5 • Find acceleration aGiven: T = 10 s, r = 25 m First, find w in rad/s Then, find acceleration = 9.87 m/s2 = 1.006 g

Solution of Ex.5 b) Find the tension Given m = 1000 kg, a = 1.006 g = 9870 N

Example 6. • A race car speeds around a circular track. • If the coefficient of friction with the tires is 1.1, what is the maximum centripetal acceleration (in “g”s) that the race car can experience? • a) Find the maximum centripetal acceleration Known: m = 1.1Remember, only consider forces towards center Maximum a = 1.1 g

Solution of Ex.6(b) b) What is the minimum circumference of the track that would permit the race car to travel at 300 km/hr? b) Find the minumum circumference Known: v = 300 km/hr = 83.33 m/s, a = 1.1 g First, find radius Then, find circumference = 4.043 m In the real world: tracks are banked

Example 7. A yo-yo is spun in a circle as shown. If the length of the string is L = 35 cm and the circular path is repeated 1.5 times per second, at what angle q (with respect to the vertical) does the string bend?

Solution of Ex.7 Apply F=ma for both the horizontal and vertical components. r = Lsinq

Solution of Ex.7 We want to find q, given w=2p·1.5 & L=0.35 2 eq.s & 2 unknowns (T and q) q = 71 degrees

HOMEWORK Giancoli, Chapter 10 4, 12, 19, 22, 26, 27, 30, 36, 41, 46 • References • “PhysicsForScientists &Engineers with Modern Physics” Giancoli 4thedition, • PearsonInternationalEdition

Physics for Scientists & Engineers, with Modern Physics, 4 th edition Giancoli