Physics 106 Mechanics Lecture 02 Wenda Cao NJIT

Physics 106: Mechanics Lecture 02 Wenda Cao NJIT Physics Department

Rotational Equilibrium and Rotational Dynamics Rotational Kinetic Energy q Moment of Inertia q Torque q Angular acceleration q Newton 2 nd Law for Rotational Motion: Torque and angular acceleration q April 7, 2009

Rotational Kinetic Energy q An object rotating about z axis with an angular speed, ω, has rotational kinetic energy q Each particle has a kinetic energy of n K i = ½ m i vi 2 q Since the tangential velocity depends on the distance, r, from the axis of rotation, we can substitute vi = wri April 7, 2009

Rotational Kinetic Energy, cont q The total rotational kinetic energy of the rigid object is the sum of the energies of all its particles q Where I is called the moment of inertia April 7, 2009

Rotational Kinetic Energy, final q There is an analogy between the kinetic energies associated with linear motion (K = ½ mv 2) and the kinetic energy associated with rotational motion (KR= ½ Iw 2) q Rotational kinetic energy is not a new type of energy, the form is different because it is applied to a rotating object q Units of rotational kinetic energy are Joules (J) April 7, 2009

Moment of Inertia of Point Mass q For a single particle, the definition of moment of inertia is n n m is the mass of the single particle r is the rotational radius q SI units of moment of inertia are kg. m 2 q Moment of inertia and mass of an object are different quantities q It depends on both the quantity of matter and its distribution (through the r 2 term) April 7, 2009

Moment of Inertia of Point Mass q For a composite particle, the definition of moment of inertia is n n mi is the mass of the ith single particle ri is the rotational radius of ith particle SI units of moment of inertia are kg. m 2 q Consider an unusual baton made up of four sphere fastened to the ends of very light rods q Find I about an axis perpendicular to the page and passing through the point O where the rods cross q April 7, 2009

The Baton Twirler Consider an unusual baton made up of four sphere fastened to the ends of very light rods. Each rod is 1. 0 m long (a = b = 1. 0 m). M = 0. 3 kg and m = 0. 2 kg. q (a) Find I about an axis perpendicular to the page and passing through the point where the rods cross. Find KR if angular speed is q (b) The majorette tries spinning her strange baton about the axis y, calculate I of the baton about this axis and KR if angular speed is q April 7, 2009

Moment of Inertia of Extended Objects Divided the extended objects into many small volume elements, each of mass Dmi q We can rewrite the expression for I in terms of Dm q q With the small volume segment assumption, q If r is constant, the integral can be evaluated with known geometry, otherwise its variation with position must be known April 7, 2009

Moment of Inertia of a Uniform Rigid Rod q The shaded area has a mass n dm = l dx q Then the moment of inertia is April 7, 2009

Parallel-Axis Theorem In the previous examples, the axis of rotation coincided with the axis of symmetry of the object q For an arbitrary axis, the parallel-axis theorem often simplifies calculations q The theorem states I = ICM + MD 2 q n n n I is about any axis parallel to the axis through the center of mass of the object ICM is about the axis through the center of mass D is the distance from the center of mass axis to the arbitrary axis April 7, 2009

Moment of Inertia of a Uniform Rigid Rod q The moment of inertia about y is q The moment of inertia about y’ is April 7, 2009

Moment of Inertia for some other common shapes April 7, 2009

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Force vs. Torque Forces cause accelerations q What cause angular accelerations ? q A door is free to rotate about an axis through O q There are three factors that determine the effectiveness of the force in opening the door: q n n n The magnitude of the force The position of the application of the force The angle at which the force is applied April 7, 2009

Torque Definition q Torque, , is the tendency of a force to rotate an object about some axis q Let F be a force acting on an object, and let r be a position vector from a rotational center to the point of application of the force, with F perpendicular to r. The magnitude of the torque is given by April 7, 2009

Torque Units and Direction q The SI units of torque are N. m q Torque is a vector quantity q Torque magnitude is given by q Torque will have direction n If the turning tendency of the force is counterclockwise, the torque will be positive n If the turning tendency is clockwise, the torque will be negative April 7, 2009

Net Torque q The force will tend to cause a counterclockwise rotation about O q The force will tend to cause a clockwise rotation about O q S = 1 + 2 = F 1 d 1 – F 2 d 2 q If S 0, starts rotating q Rate of rotation of an q If S = 0, rotation rate object does not change, does not change unless the object is acted on by a net torque April 7, 2009

General Definition of Torque The applied force is not always perpendicular to the position vector q The component of the force perpendicular to the object will cause it to rotate q When the force is parallel to the position vector, no rotation occurs q When the force is at some angle, the perpendicular component causes the rotation q April 7, 2009

General Definition of Torque q Let F be a force acting on an object, and let r be a position vector from a rotational center to the point of application of the force. The magnitude of the torque is given by q = 0° or = 180 °: torque are equal to zero q = 90° or = 270 °: magnitude of torque attain to the maximum April 7, 2009

Understand sinθ q The component of the force (F cos ) has no tendency to produce a rotation q The moment arm, d, is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force d = r sin April 7, 2009

The Swinging Door q Two forces are applied to the door, as shown in figure. Suppose a wedge is placed 1. 5 m from the hinges on the other side of the door. What minimum force must the wedge exert so that the force applied won’t open the door? Assume F 1 = 150 N, F 2 = 300 N, F 3 = 300 N, θ = 30° F 2 F 3 2. 0 m θ F 1 April 7, 2009

Torque on a Rotating Object q q q Consider a particle of mass m rotating in a circle of radius r under the influence of tangential force The tangential force provides a tangential acceleration: Ft = mat Multiply both side by r, then r. Ft = mrat Since at = r , we have r. Ft = mr 2 So, we can rewrite it as = mr 2 = I April 7, 2009

Torque on a Solid Disk q q q Consider a solid disk rotating about its axis. The disk consists of many particles at various distance from the axis of rotation. The torque on each one is given by = mr 2 The net torque on the disk is given by = ( mr 2) A constant of proportionality is the moment of inertia, I = mr 2 = m 1 r 12 + m 2 r 22 + m 3 r 32 + … So, we can rewrite it as = I April 7, 2009

Newton’s Second Law for a Rotating Object q When a rigid object is subject to a net torque (≠ 0), it undergoes an angular acceleration The angular acceleration is directly proportional to the net torque q The angular acceleration is inversely proportional to the moment of inertia of the object q The relationship is analogous to q April 7, 2009

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The Falling Object A solid, frictionless cylindrical reel of mass M = 3. 0 kg and radius R = 0. 4 m is used to draw water from a well. A bucket of mass m = 2. 0 kg is attached to a cord that is wrapped around the cylinder. q (a) Find the tension T in the cord and acceleration a of the object. q (b) If the object starts from rest at the top of the well and falls for 3. 0 s before hitting the water, how far does it fall ? q April 7, 2009

Example, Newton’s Second Law for Rotation Draw free body diagrams of each object q Only the cylinder is rotating, so apply St = I a q The bucket is falling, but not rotating, so apply SF = ma q Remember that a = a r and solve the resulting equations q April 7, 2009