ANGULAR ACCELERATION and MOMENT of INERTIA Introduction
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Rotational Motion (The Dynamics of a Rigid Body)
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Robert Katz Publications Research Papers in Physics and Astronomy 1-1958 Physics, Chapter 11: Rotational Motion (The Dynamics of a Rigid Body) Henry Semat City College of New York Robert Katz University of Nebraska-Lincoln, [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/physicskatz Part of the Physics Commons Semat, Henry and Katz, Robert, "Physics, Chapter 11: Rotational Motion (The Dynamics of a Rigid Body)" (1958). Robert Katz Publications. 141. https://digitalcommons.unl.edu/physicskatz/141 This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Robert Katz Publications by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. 11 Rotational Motion (The Dynamics of a Rigid Body) 11-1 Motion about a Fixed Axis The motion of the flywheel of an engine and of a pulley on its axle are examples of an important type of motion of a rigid body, that of the motion of rotation about a fixed axis. Consider the motion of a uniform disk rotat ing about a fixed axis passing through its center of gravity C perpendicular to the face of the disk, as shown in Figure 11-1. The motion of this disk may be de scribed in terms of the motions of each of its individual particles, but a better way to describe the motion is in terms of the angle through which the disk rotates. -
M1=100 Kg Adult, M2=10 Kg Baby. the Seesaw Starts from Rest. Which Direction Will It Rotates?
m1 m2 m1=100 kg adult, m2=10 kg baby. The seesaw starts from rest. Which direction will it rotates? (a) Counter-Clockwise (b) Clockwise ()(c) NttiNo rotation (d) Not enough information Effect of a Constant Net Torque 2.3 A constant non-zero net torque is exerted on a wheel. Which of the following quantities must be changing? 1. angular position 2. angular velocity 3. angular acceleration 4. moment of inertia 5. kinetic energy 6. the mass center location A. 1, 2, 3 B. 4, 5, 6 C. 1,2, 5 D. 1, 2, 3, 4 E. 2, 3, 5 1 Example: second law for rotation PP10601-50: A torque of 32.0 N·m on a certain wheel causes an angular acceleration of 25.0 rad/s2. What is the wheel's rotational inertia? Second Law example: α for an unbalanced bar Bar is massless and originally horizontal Rotation axis at fulcrum point L1 N L2 Î N has zero torque +y Find angular acceleration of bar and the linear m1gmfulcrum 2g acceleration of m1 just after you let go τnet Constraints: Use: τnet = Itotα ⇒ α = Itot 2 2 Using specific numbers: where: Itot = I1 + I2 = m1L1 + m2L2 Let m1 = m2= m L =20 cm, L = 80 cm τnet = ∑ τo,i = + m1gL1 − m2gL2 1 2 θ gL1 − gL2 g(0.2 - 0.8) What happened to sin( ) in moment arm? α = 2 2 = 2 2 L1 + L2 0.2 + 0.8 2 net = − 8.65 rad/s Clockwise torque m gL − m gL a ==+ -α L 1.7 m/s2 α = 1 1 2 2 11 2 2 Accelerates UP m1L1 + m2L2 total I about pivot What if bar is not horizontal? 2 See Saw 3.1. -
Chapter 3 Motion in Two and Three Dimensions
Chapter 3 Motion in Two and Three Dimensions 3.1 The Important Stuff 3.1.1 Position In three dimensions, the location of a particle is specified by its location vector, r: r = xi + yj + zk (3.1) If during a time interval ∆t the position vector of the particle changes from r1 to r2, the displacement ∆r for that time interval is ∆r = r1 − r2 (3.2) = (x2 − x1)i +(y2 − y1)j +(z2 − z1)k (3.3) 3.1.2 Velocity If a particle moves through a displacement ∆r in a time interval ∆t then its average velocity for that interval is ∆r ∆x ∆y ∆z v = = i + j + k (3.4) ∆t ∆t ∆t ∆t As before, a more interesting quantity is the instantaneous velocity v, which is the limit of the average velocity when we shrink the time interval ∆t to zero. It is the time derivative of the position vector r: dr v = (3.5) dt d = (xi + yj + zk) (3.6) dt dx dy dz = i + j + k (3.7) dt dt dt can be written: v = vxi + vyj + vzk (3.8) 51 52 CHAPTER 3. MOTION IN TWO AND THREE DIMENSIONS where dx dy dz v = v = v = (3.9) x dt y dt z dt The instantaneous velocity v of a particle is always tangent to the path of the particle. 3.1.3 Acceleration If a particle’s velocity changes by ∆v in a time period ∆t, the average acceleration a for that period is ∆v ∆v ∆v ∆v a = = x i + y j + z k (3.10) ∆t ∆t ∆t ∆t but a much more interesting quantity is the result of shrinking the period ∆t to zero, which gives us the instantaneous acceleration, a. -
Rotational Motion of Electric Machines
Rotational Motion of Electric Machines • An electric machine rotates about a fixed axis, called the shaft, so its rotation is restricted to one angular dimension. • Relative to a given end of the machine’s shaft, the direction of counterclockwise (CCW) rotation is often assumed to be positive. • Therefore, for rotation about a fixed shaft, all the concepts are scalars. 17 Angular Position, Velocity and Acceleration • Angular position – The angle at which an object is oriented, measured from some arbitrary reference point – Unit: rad or deg – Analogy of the linear concept • Angular acceleration =d/dt of distance along a line. – The rate of change in angular • Angular velocity =d/dt velocity with respect to time – The rate of change in angular – Unit: rad/s2 position with respect to time • and >0 if the rotation is CCW – Unit: rad/s or r/min (revolutions • >0 if the absolute angular per minute or rpm for short) velocity is increasing in the CCW – Analogy of the concept of direction or decreasing in the velocity on a straight line. CW direction 18 Moment of Inertia (or Inertia) • Inertia depends on the mass and shape of the object (unit: kgm2) • A complex shape can be broken up into 2 or more of simple shapes Definition Two useful formulas mL2 m J J() RRRR22 12 3 1212 m 22 JRR()12 2 19 Torque and Change in Speed • Torque is equal to the product of the force and the perpendicular distance between the axis of rotation and the point of application of the force. T=Fr (Nm) T=0 T T=Fr • Newton’s Law of Rotation: Describes the relationship between the total torque applied to an object and its resulting angular acceleration. -
Physics B Topics Overview ∑
Physics 106 Lecture 1 Introduction to Rotation SJ 7th Ed.: Chap. 10.1 to 3 • Course Introduction • Course Rules & Assignment • TiTopics Overv iew • Rotation (rigid body) versus translation (point particle) • Rotation concepts and variables • Rotational kinematic quantities Angular position and displacement Angular velocity & acceleration • Rotation kinematics formulas for constant angular acceleration • Analogy with linear kinematics 1 Physics B Topics Overview PHYSICS A motion of point bodies COVERED: kinematics - translation dynamics ∑Fext = ma conservation laws: energy & momentum motion of “Rigid Bodies” (extended, finite size) PHYSICS B rotation + translation, more complex motions possible COVERS: rigid bodies: fixed size & shape, orientation matters kinematics of rotation dynamics ∑Fext = macm and ∑ Τext = Iα rotational modifications to energy conservation conservation laws: energy & angular momentum TOPICS: 3 weeks: rotation: ▪ angular versions of kinematics & second law ▪ angular momentum ▪ equilibrium 2 weeks: gravitation, oscillations, fluids 1 Angular variables: language for describing rotation Measure angles in radians simple rotation formulas Definition: 360o 180o • 2π radians = full circle 1 radian = = = 57.3o • 1 radian = angle that cuts off arc length s = radius r 2π π s arc length ≡ s = r θ (in radians) θ ≡ rad r r s θ’ θ Example: r = 10 cm, θ = 100 radians Æ s = 1000 cm = 10 m. Rigid body rotation: angular displacement and arc length Angular displacement is the angle an object (rigid body) rotates through during some time interval..... ...also the angle that a reference line fixed in a body sweeps out A rigid body rotates about some rotation axis – a line located somewhere in or near it, pointing in some y direction in space • One polar coordinate θ specifies position of the whole body about this rotation axis. -
6. Non-Inertial Frames
6. Non-Inertial Frames We stated, long ago, that inertial frames provide the setting for Newtonian mechanics. But what if you, one day, find yourself in a frame that is not inertial? For example, suppose that every 24 hours you happen to spin around an axis which is 2500 miles away. What would you feel? Or what if every year you spin around an axis 36 million miles away? Would that have any e↵ect on your everyday life? In this section we will discuss what Newton’s equations of motion look like in non- inertial frames. Just as there are many ways that an animal can be not a dog, so there are many ways in which a reference frame can be non-inertial. Here we will just consider one type: reference frames that rotate. We’ll start with some basic concepts. 6.1 Rotating Frames Let’s start with the inertial frame S drawn in the figure z=z with coordinate axes x, y and z.Ourgoalistounderstand the motion of particles as seen in a non-inertial frame S0, with axes x , y and z , which is rotating with respect to S. 0 0 0 y y We’ll denote the angle between the x-axis of S and the x0- axis of S as ✓.SinceS is rotating, we clearly have ✓ = ✓(t) x 0 0 θ and ✓˙ =0. 6 x Our first task is to find a way to describe the rotation of Figure 31: the axes. For this, we can use the angular velocity vector ! that we introduced in the last section to describe the motion of particles. -
Rotation: Moment of Inertia and Torque
Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn that where the force is applied and how the force is applied is just as important as how much force is applied when we want to make something rotate. This tutorial discusses the dynamics of an object rotating about a fixed axis and introduces the concepts of torque and moment of inertia. These concepts allows us to get a better understanding of why pushing a door towards its hinges is not very a very effective way to make it open, why using a longer wrench makes it easier to loosen a tight bolt, etc. This module begins by looking at the kinetic energy of rotation and by defining a quantity known as the moment of inertia which is the rotational analog of mass. Then it proceeds to discuss the quantity called torque which is the rotational analog of force and is the physical quantity that is required to changed an object's state of rotational motion. Moment of Inertia Kinetic Energy of Rotation Consider a rigid object rotating about a fixed axis at a certain angular velocity. Since every particle in the object is moving, every particle has kinetic energy. To find the total kinetic energy related to the rotation of the body, the sum of the kinetic energy of every particle due to the rotational motion is taken. The total kinetic energy can be expressed as .. -
Lecture 25 Torque
Physics 170 - Mechanics Lecture 25 Torque Torque From experience, we know that the same force will be much more effective at rotating an object such as a nut or a door if our hand is not too close to the axis. This is why we have long-handled wrenches, and why the doorknobs are not next to the hinges. Torque We define a quantity called torque: The torque increases as the force increases and also as the distance increases. Torque The ability of a force to cause a rotation or twisting motion depends on three factors: 1. The magnitude F of the force; 2. The distance r from the point of application to the pivot; 3. The angle φ at which the force is applied. = r x F Torque This leads to a more general definition of torque: Two Interpretations of Torque Torque Only the tangential component of force causes a torque: Sign of Torque If the torque causes a counterclockwise angular acceleration, it is positive; if it causes a clockwise angular acceleration, it is negative. Sign of Torque Question Five different forces are applied to the same rod, which pivots around the black dot. Which force produces the smallest torque about the pivot? (a) (b) (c) (d) (e) Gravitational Torque An object fixed on a pivot (taken as the origin) will experience gravitational forces that will produce torques. The torque about the pivot from the ith particle will be τi=−ximig. The minus sign is because particles to the right of the origin (x positive) will produce clockwise (negative) torques. -
Newton's First Law of Motion the Principle of Relativity
MATHEMATICS 7302 (Analytical Dynamics) YEAR 2017–2018, TERM 2 HANDOUT #1: NEWTON’S FIRST LAW AND THE PRINCIPLE OF RELATIVITY Newton’s First Law of Motion Our experience seems to teach us that the “natural” state of all objects is at rest (i.e. zero velocity), and that objects move (i.e. have a nonzero velocity) only when forces are being exerted on them. Aristotle (384 BCE – 322 BCE) thought so, and many (but not all) philosophers and scientists agreed with him for nearly two thousand years. It was Galileo Galilei (1564–1642) who first realized, through a combination of experi- mentation and theoretical reflection, that our everyday belief is utterly wrong: it is an illusion caused by the fact that we live in a world dominated by friction. By using lubricants to re- duce friction to smaller and smaller values, Galileo showed experimentally that objects tend to maintain nearly their initial velocity — whatever that velocity may be — for longer and longer times. He then guessed that, in the idealized situation in which friction is completely eliminated, an object would move forever at whatever velocity it initially had. Thus, an object initially at rest would stay at rest, but an object initially moving at 100 m/s east (for example) would continue moving forever at 100 m/s east. In other words, Galileo guessed: An isolated object (i.e. one subject to no forces from other objects) moves at constant velocity, i.e. in a straight line at constant speed. Any constant velocity is as good as any other. This principle was later incorporated in the physical theory of Isaac Newton (1642–1727); it is nowadays known as Newton’s first law of motion. -
Rotational Motion and Angular Momentum 317
CHAPTER 10 | ROTATIONAL MOTION AND ANGULAR MOMENTUM 317 10 ROTATIONAL MOTION AND ANGULAR MOMENTUM Figure 10.1 The mention of a tornado conjures up images of raw destructive power. Tornadoes blow houses away as if they were made of paper and have been known to pierce tree trunks with pieces of straw. They descend from clouds in funnel-like shapes that spin violently, particularly at the bottom where they are most narrow, producing winds as high as 500 km/h. (credit: Daphne Zaras, U.S. National Oceanic and Atmospheric Administration) Learning Objectives 10.1. Angular Acceleration • Describe uniform circular motion. • Explain non-uniform circular motion. • Calculate angular acceleration of an object. • Observe the link between linear and angular acceleration. 10.2. Kinematics of Rotational Motion • Observe the kinematics of rotational motion. • Derive rotational kinematic equations. • Evaluate problem solving strategies for rotational kinematics. 10.3. Dynamics of Rotational Motion: Rotational Inertia • Understand the relationship between force, mass and acceleration. • Study the turning effect of force. • Study the analogy between force and torque, mass and moment of inertia, and linear acceleration and angular acceleration. 10.4. Rotational Kinetic Energy: Work and Energy Revisited • Derive the equation for rotational work. • Calculate rotational kinetic energy. • Demonstrate the Law of Conservation of Energy. 10.5. Angular Momentum and Its Conservation • Understand the analogy between angular momentum and linear momentum. • Observe the relationship between torque and angular momentum. • Apply the law of conservation of angular momentum. 10.6. Collisions of Extended Bodies in Two Dimensions • Observe collisions of extended bodies in two dimensions. • Examine collision at the point of percussion. -
Vectors and Coordinate Systems
Vectors and Coordinate Systems In Newtonian mechanics, we want to understand how material bodies interact with each other and how this affects their motion through space. In order to be able to make quantitative statements about this, we need a mathematical language for describing motion, which is known as kinematics. Part of the language of kinematics involves calculus, since we know that calculus lets us talk about how things change in a quantitative way. The other part of this mathematical language involves the notion of a vector, and the related concept of a coordinate system. Today we'll review the basic concepts of vectors and coordinate systems that we'll need, and then we'll add ideas from calculus in order to form the subject of kinematics. The starting point for describing the location of objects in space is to pick an arbitrary location, known as the origin, and to describe every other position in space by the displacement vector between the origin and that point. This is shown in Figure 1 Because vectors allow us to talk about lengths and orientations in a quantitative way, they are a natural tool for describing motion. Much in the same way that I might say my house is 1.3 miles Northwest of campus, I can say that the location of some particle is a certain distance from the origin, at some angle. We won't belabour all of the basic properties of vectors (such as addition and subtraction), but if you are rusty on these subjects, now is the time to review! Figure 1: A point in space being indicated by its displacement vector from the origin. -
0230 Lecture Notes - a Tale of Three Accelerations Or the Differences Between Angular, Tangential, and Centripetal Accelerations.Docx Page 1 of 1
Flipping Physics Lecture Notes: A Tale of Three Accelerations or The Differences between Angular, Tangential, and Centripetal Accelerations https://www.flippingphysics.com/3-accelerations.html An object moving in a circle can have three different types of accelerations: Δω rad • Angular Acceleration: in is an angular quantity. α = 2 Δt s m • Tangential Acceleration: in is a linear quantity. at = rα 2 s v 2 m • Centripetal Acceleration: t 2 in is a linear quantity. ac = = rω 2 r s Angular acceleration separates itself from the others: 1) Because it is an angular quantity, whereas the other two are linear quantities. 2) Because angular acceleration applies to the whole rigid object, however, tangential acceleration and centripetal acceleration are for a specific radius. A major difference between tangential acceleration and centripetal acceleration is their direction. • Centripetal means “center seeking”. Centripetal acceleration is always directed inward. • Tangential acceleration is always directed tangent to the circle. o By definition, tangential acceleration and centripetal acceleration are perpendicular to one another. Another major difference between tangential acceleration and centripetal acceleration is that circular motion cannot exist without centripetal acceleration. • No centripetal acceleration means the object is not moving in a circle. o Centripetal acceleration results from the change in direction of the tangential velocity. If the tangential velocity is not changing directions, then the object is not moving in a circle. • Tangential acceleration results from the change in magnitude of the tangential velocity of an object. An object can move in a circle and not have any tangential acceleration. No tangential acceleration simply means the angular acceleration of the object is zero and the object is moving with a constant angular velocity.