This document discusses rotational motion and provides definitions and equations for key angular quantities such as angular displacement (θ), angular velocity (ω), angular acceleration (α), torque (τ), moment of inertia (I), angular momentum (L), and rotational kinetic energy. It defines these quantities, gives their relationships to linear motion quantities, and provides examples of how to set up and solve problems involving rotational dynamics.
The Poynting theorem represents the time rate change of electromagnetic energy within a certain volume plus the time rate of energy flowing out through the boundary surface is equal to the power transferred into the electromagnetic field.
This statement follows the conservation of energy in electromagnetism and is known as the Poynting theorem.
Cylindrical and spherical coordinates shalinishalini singh
In this Presentation, I have explained the co-ordinate system in three plain. ie Cylindrical, Spherical, Cartesian(Rectangular) along with its Differential formulas for length, area &volume.
The Poynting theorem represents the time rate change of electromagnetic energy within a certain volume plus the time rate of energy flowing out through the boundary surface is equal to the power transferred into the electromagnetic field.
This statement follows the conservation of energy in electromagnetism and is known as the Poynting theorem.
Cylindrical and spherical coordinates shalinishalini singh
In this Presentation, I have explained the co-ordinate system in three plain. ie Cylindrical, Spherical, Cartesian(Rectangular) along with its Differential formulas for length, area &volume.
this is class 12 Maharashtra board physics subject content. this is complete content with notes with easily explaination.
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This ppt is as per class 12 Maharashtra State Board's new syllabus w.e.f. 2020. Images are taken from Google public sources and Maharashtra state board textbook of physics. Gif(videos) from Giphy.com. Only intention behind uploading these ppts is to help state board's class 12 students understand physics concepts.
Introduction to the structure of atoms from the view of a chemist - what are neutrons protons and electrons and how are they organized ? How are electrons organized - in 3 quantum numbers. Experimental evidence from the Bohr model.
Dealing with Notations and conventions in tensor analysis-Einstein's summation convention covariant and contravariant and mixed tensors-algebraic operations in tensor symmetric and skew symmetric tensors-tensor calculus-Christoffel symbols-kinematics in Riemann space-Riemann-Christoffel tensor.
this is class 12 Maharashtra board physics subject content. this is complete content with notes with easily explaination.
for buying or neet attractive ppt in any subject contact me 8879919898. go to my site akchem.tk
blog akchem.blogspot.com
This ppt is as per class 12 Maharashtra State Board's new syllabus w.e.f. 2020. Images are taken from Google public sources and Maharashtra state board textbook of physics. Gif(videos) from Giphy.com. Only intention behind uploading these ppts is to help state board's class 12 students understand physics concepts.
Introduction to the structure of atoms from the view of a chemist - what are neutrons protons and electrons and how are they organized ? How are electrons organized - in 3 quantum numbers. Experimental evidence from the Bohr model.
Dealing with Notations and conventions in tensor analysis-Einstein's summation convention covariant and contravariant and mixed tensors-algebraic operations in tensor symmetric and skew symmetric tensors-tensor calculus-Christoffel symbols-kinematics in Riemann space-Riemann-Christoffel tensor.
Discuss the law of universal gravitation and satellite motion.
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Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
3. 8-1 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 . When P (at radius r) travels an arc length , OP sweeps out an angle θ . θ Angular Displacement of the body
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6. θ 3 10 -4 rad = ? º r = 100 m, = ? a) θ = (3 10 -4 rad) [(360/2 π )º/rad] = 0.017º For small angles: The chord arc length b) = r θ = (100) (3 10 -4 ) = 0.03 m = 3 cm θ MUST be in radians in part b Example 8-2 :
7. 8-1 Angular Quantities Angular displacement: The average angular velocity is defined as the total angular displacement divided by time : The instantaneous angular velocity: (8-2a) (8-2b) (Units = rad/s ), Valid ONLY if θ is in rad !
8. 8-1 Angular Quantities The angular acceleration is the rate at which the angular velocity changes with time: The instantaneous acceleration: (8-3a) (8-3b) (Units = rad/s 2 ) Valid ONLY if θ is in rad & ω is in rad/s !
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10. Connection Between Angular & Linear Quantities v = ( / t), = r θ v = r( θ / t) = r ω Radians! v = r ω Depends on r ( ω is the same for all points!) v A = r A ω A v B = r B ω B v B > v A since r B > r A Therefore, objects farther from the axis of rotation will move faster .
11. 8-1 Angular Quantities If the angular velocity of a rotating object changes , it has a tangential acceleration: a tan = ( v/ t), v =r ω = r ( ω / t) a tan = r α Even if the angular velocity is constant, each point on the object has a centripetal acceleration:
12. 8-1 Angular Quantities Here is the correspondence between linear and rotational quantities:
13. 8-1 Angular Quantities The frequency is the number of complete revolutions per second: or ω = 2 π f (angular frequency) Frequencies are measured in hertz . The period is the time one revolution takes:
14. 8-2 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. NOTE: These are ONLY VALID if all angular quantities are in radian units!!
15. 8-3 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 reference frame 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 :
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17. 8-4 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 .
18. 8-4 Torque A longer lever arm is very helpful in rotating objects.
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20. 8-4 Torque Here, the lever arm for F A is the distance from the knob to the hinge ; the lever arm for F D is zero ; and the lever arm for F C is as shown.
21. 8-4 Torque The torque is defined as: F = F sin θ F = F cos θ τ = rF sin θ Units of torque: Newton-meters (N m)
22. Example 8-9 r = r 2 sin60 º τ 2 = -r 2 F 2 sin60 º τ 1 = r 1 F 1 τ = τ 1 + τ 2 = -6.7 m N Always use the following sign convention ! Counterclockwise rotation + torque Clockwise rotation - torque
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24. Simplest Possible Case A mass m moving in a Circle of radius r , one force F TANGENTIAL to the circle τ = rF Newton’s 2 nd Law + relation (a = r α ) between tangential & angular accelerations F = ma = mr α So τ = mr 2 α Newton’s 2 nd Law for Rotations Proportionality constant between τ & α is mr 2 (point mass only!)
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27. 8-5 Rotational Dynamics; Torque and Rotational Inertia The quantity is called the rotational inertia 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.
28. 8-5 Rotational Dynamics; Torque and Rotational Inertia The rotational inertia of an object depends not only on its mass distribution but also the location of the axis of rotation – compare (f) and (g), for example.
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31. 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. 8-6 Solving Problems in Rotational Dynamics
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33. 8-7 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: An object that has both translational and rotational motion also has both translational and rotational kinetic energy : (8-15) (8-16)
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35. 8-7 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.
36. 8-7 Rotational Kinetic Energy The torque does work as it moves the wheel through an angle θ : (8-17) Torque: τ = Fr Work: W = F = Fr = τ
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38. 8-8 Angular Momentum and Its Conservation Therefore, systems that can change their rotational inertia through internal forces will also change their rate of rotation: