Calculating rotational inertia. Show development of expressions.

In the case of linear motion, the rotational inertia is known as an analog of mass. For a rotary load, it's the product of its mass and the square of the perpendicular distance of the mass from the axis. Step 4: Calculate the desired unknown value. Strategy. In order to understand rotational inertia, we should first review the equation for rotational inertia of a system of particles: The rotational inertia of a system of particles equals the sum of the quantity of the mass of each particle times the Feb 29, 2024 · The product of inertia Ixy of a rectangle is zero, because x and y are symmetry axes. Rotational mechanics; Moment of inertia; Energy; Calculator Apps. Calculate the rotational inertia of a wheel that has a kinetic energy of 24,400 J when rotating at 676 rev/min. Calculating the moment of inertia of a rod about its center of mass is a good example of the need for calculus to deal with the properties of continuous mass distributions. It is a quantity which describes the relationship between an object's angular momunetum and it's angular velocity. In physical terms, it could be percieved as a measure of how "difficult" it is to rotate an object at a given angular velocity, and is derived from Sep 27, 2020 · Introduction. The work in rotational motion is J. For mass M = kg. Assembly Stack-up Analysis. I = 1 12 × m b × L 2. In conclusion, the Moment of Inertia Calculator is an essential computational tool that rapidly and precisely generates moment of inertia values for objects of various shapes and sizes. e. The moments of inertia about principal Jun 21, 2024 · In this situation, the angular momentum is the product of the moment of inertia, \text I I, and the angular velocity, \text ω ω. I (solid sphere) = kg m 2. Figure 10. It is Nov 2, 2018 · Calculating rotational inertia is important in understanding an object's behavior when rotating. Feb 4, 2024 · I is the rotational inertia; m is the mass of the rotating object; r is the radius of the circular path; Sample Question. Now, in the case of non-uniform objects, we can calculate the moment Calculating Rotational Inertia for Basic Geometric Objects. Jul 20, 2022 · Moment of Inertia of a Rod of Uniform Mass Density. The moment of inertia of a point mass is given by I = mr 2 , but the rod would have to be considered to be an infinite number of point masses, and each must be multiplied by Rotational Inertia or Moment of Inertia. Dynamics is concerned with force and mass and their effects on motion. ) kg m 2 Moment of inertia: Also known as the rotational inertia, of an object is a quantity that determines the torque needed to achieve a desired angular acceleration about a rotational axis. 2: An object of mass 3kg is rotating a circular path. Newton: I think about it from the basic formula F = mass times acceleration (F=ma). (Treat the stick as a thin rod. A known torque is applied to the pulley on the Rotary Motion Sensor, causing a disk and ring to rotate. and radius R = cm. Jan 30, 2008 · Rotational inertia plays a crucial role in the motion of a pendulum. 5 * Mass (m) * Velocity squared (v2) Where: – Mass (m) is the mass of the object in kilograms. Rotational inertia is a measure of the resistance of an object to changes in its angular velocity. Solution: We have, m = 20. Fundamental Vehicle Dynamics. Our moment of inertia calculator can be of assistance if Introduction to Moment of Inertia. However, we know how to integrate over space, not over mass. Problem 56P: Calculate the rotational kinetic energy of a 12-kg motorcycle wheel if its angular velocity is 120 Jun 21, 2023 · How To Calculate Moment of Inertia. A particle of mass m at radius r from the axis of rotation moves in a circle of radius r with an angular speed Ω about this axis. , the opposition that the body exhibits to having its speed of rotation about an axis altered by the application of a torque (turning force). (Treat the stick as a thin rod I= 1/12ML 2. Dec 28, 2020 · Different objects have different moments of inertia. It is having rotational inertia of 300 Kg m². This states that the rotational inertia of an object can be calculated by adding the rotational inertia of each individual component, taking into account the distance of each component from the axis of rotation. The quantity mr2 is called the rotational inertia or moment of inertia of a point Explain how the moment of inertia of rigid bodies affects their rotational kinetic energy; Use conservation of mechanical energy to analyze systems undergoing both rotation and translation; Calculate the angular velocity of a rotating system when there are energy losses due to nonconservative forces Moment of Inertia formula: I = m × r2. It is the prime parameter that keeps an object under rotatory motion. While applying the torque, measure the angular acceleration. This angular velocity calculator finds angular velocity in two ways. You can either use the angle change in the period of time or apply the linear A force F is applied to the object perpendicular to the radius r, causing it to accelerate about the pivot point. Imagine applying a known torque to an object. How does this change affect the amount of blue putty needed at the number 3 to just rotate the circle? Change the circle’s moment of inertia and then try rotating the circle by using different amounts of blue putty. Entering ω and I into the expression for rotational kinetic energy gives. \ [I = \sum_ {i} m_i r_i^2 \] for all the point masses that make up the object. Both are a measure of how much an object resists Rotational inertia or moment of inertia, is the mathematical way to describe a n object's reluctance to changes in its rotational motion. It represents the rotational inertia of an object, which is the measure of an object’s resistance to changes in its rotation about a particular axis. Apr 9, 2015 · 097 - Rotational InertiaIn this video Paul Andersen explains how the angular momentum of an object if a product of the rotational inertia and the angular vel Moment of Inertia is a physical quantity that resist to transitional motion. So up you have: N. The moment of inertia integral is an integral over the mass distribution. The angular momentum in rotational motion is kg∙m2/s. k = length (radius of gyration) (ft) or any other unit of length. 5 * m * r^2, where I is the rotational inertia, m is the mass of the spool, and r is the radius of the spool. Generally, for uniform objects, the moment of inertia is calculated by taking the square of its distance from the axis of rotation (r 2) and the product of its mass. The Moment of Inertia of a point object rotating around a given axis is kg∙m 2. An obsolete term for rotational inertia is "moment of inertia," a term sometimes found in Quest problems. What is the unit for rotational inertia? The unit for rotational inertia is kilogram-meter squared (kg·m^2). E. Dynamics of Rotational Motion Calculator Results (detailed calculations and formula below) The torque calculated by applying Newton's Second Law in the Rotational Motion is N×m. 0 m/s, and compare it with the rotational energy in the blades. It helps in designing machines and structures to ensure stability and proper functioning. (a) Calculate the rotational kinetic energy in the blades when they rotate at 300 rpm. 3 s (without the cord slipping on the pulley). 2: An object is supported by a horizontal frictionless table and is attached to a pivot point by a cord that supplies centripetal force. Figure 6. Solution: Feb 13, 2024 · calculation for Moment of Inertia; Inertia Forces calculation for Mechanical Engineering; rotary engine geometry calculation for Calculations; Explore. where: I: Moment of Inertia, in kg. = 20 ×times 7². Rotational Inertia = m(r)(r), where "m" is the mass and "r" is the radius or the distance between the object and the axis. r = 4. The greater the rotational inertia, the more energy is required to change the object's rotational motion. 0kg ⋅ m2. The moment of inertia ( I ), however, is Jun 17, 2019 · The moment of inertia is the quantitative measure of rotational inertia, just as in translational motion, and mass is the quantitative measure of linear inertia—that is, the more massive an object is, the more inertia it has, and the greater is its resistance to change in linear velocity. The Moment of Inertia, often symbolized as I, is a fundamental concept in physics and engineering, particularly in the fields of mechanics and structural analysis. Apr 24, 2017 · Rotational inertia is calculated for objects rotating about an axis. M = mass (slug) or other correct unit of mass. Rotational Inertia Calculator. It represents how difficult it overcomed to change its angular motion about that axis. r^2, so unit is kg. Sep 12, 2022 · The moment of inertia of one blade is that of a thin rod rotated about its end, listed in Figure 10. We define dm to be a small element of mass making up the rod. Moment of inertia is larger when an object’s mass is farther from the axis of rotation. Approximate the nitrogen molecule as a rigid dumbbell of two equal point masses and calculate the moment of inertia about its center of mass. The smaller the resulting angular acceleration, the larger the object’s rotational inertia. Question 1. File Catalog. We don't yet know the angular acceleration, alpha, but we do know the initial angular Apr 27, 2022 · Load inertia, or moment of inertia, is the resistance of any physical object to any change in its speed from the perspective of the rotational axis . Rotating unbalance. Oct 26, 2020 · Two different mass moments of inertia. Dynamics for rotational motion is completely analogous to linear or translational dynamics. Rotational inertia is a scalar, not a vector and is dependent upon the radius of rotation according to the formula rotational inertia = mass x radius^2. Dec 5, 2007 · To calculate rotational inertia for multiple masses and rods, you can use the parallel axis theorem. [1] In the angular version of Newton’s 2nd law, torque τ takes the place of force and rotational inertia takes the place of mass. The next part of my lab involves calculating the experimental rotational inertia of the ring and disk, and I know the hanging mass, slop and radius of each one. 0kg)(4. It is a scalar value that considers the mass distribution around the axis and serves as the rotational analog of mass in linear motion. Show development of expressions. A rotating mass, or rotor, is said to be out of balance when its center of mass (inertia axis) is out of alignment with the center of rotation (geometric axis). Moment of Inertia: This term is a measure of an object's resistance to rotational motion about a given axis. (Similar to the way inertia describes an object's reluctance to change in its linear motion. The moment of inertia about a given point does depend on how that mass is distributed about the point / axis you are calculating the moment of inertia about. May 21, 2023 · Calculating Rotational Inertia for Continuous Objects Our task is to compute the rotational inertia, for which the formula in terms of masses and their positions is different from the one for center of mass (see Section 4. Firstly: Make sure all your parts have the correct material defined (with correct material densities). Mathematically torque is given by: The distance from the axle, or axis of rotation, of the four masses on the spokes can be adjusted. The torque applied to an object begins to rotate it with an acceleration inversely proportional to its moment of inertia. The force is kept perpendicular to r. For example, if we attach a rotating disc to a massless rope and then pull on the rope with constant force, we Calculating Moment Of Inertia. So, make sure you are using the same reference point for your rotational acceleration and mass moment of inertia. Calculating Rotational Inertia for Continuous Objects Our task is to compute the rotational inertia, for which the formula in terms of masses and their positions is different from the one for center of mass (see Section 4. 269 kg, about an axis perpendicular to the stick and located at the 16. What is the equation for calculating the speed of a rotating square plate? Jan 30, 2023 · Rotation of Linear Molecules. 2 ), but the procedure is exactly the same. In principal axes, that are rotated by an angle θ relative to original centroidal ones x,y, the product of inertia becomes zero. Unbalance causes a moment which gives the rotor a wobbling movement characteristic This calculator computes the inertia (also known as the moment of inertia) of a rotating object using the following formula: Inertia (I) = Mass (m) * Radius squared (r2) + 0. Solid Cylinder or Disk: I = 1/2 * M * R². Q. The total I is four times this moment of inertia because there are four blades. When released from rest, block 2 falls 80 cm in 5. So, while the analogies are precise, these rotational quantities depend on more factors. , the moment of inertia, I ) of the object and its angular velocity ω ), which is measured in degrees/s or rad/s. In addition, the High School Physics Laboratory Manual addresses content in this section in the lab titled: Circular and Rotational Motion, as well as the following standards: These methods can be used to find the moment of inertia of things like spheres, hollow spheres, thin spherical shells and other more exotic shapes like cones, buckets, and eggs — basically, anything that might roll and that has a fairly simple mathematical description. C3 Project Docs. It explains how to calculate the angular momentum of a rotating object and Nov 21, 2023 · The moment of inertia (or rotational inertia),I, is 600, and we're asked to calculate the torque, tau. 8), the component of the torque about the z -axis is then given by. Moments of inertia can be found by summing or integrating over every ‘piece of mass’ that makes up an object, multiplied by the square of the distance of each ‘piece of mass’ to the axis. r: Distance from the Rotating Axis, in m. 73 kg, about an axis perpendicular to the stick and located at the 18 cm mark. From there I'd probably use τ = I α but I know the angular acceleration nor how to calculate it. If we have a rigid body rotating at an angular speed Ω about a fixed axis, each particle in it will have a certain amount of kinetic energy. The helicopter has a total loaded mass of 1000 kg. Automotive Quality. Now to answer your question, yes you can easily get the mass moment of Inertia from Inventor. The quantity mr2 m r 2 is called the rotational inertia or moment of inertia Dec 7, 2004 · How do you calculate rotational inertia of a sphere? The formula for calculating the rotational inertia of a solid sphere is I = (2/5) * m * r^2, where I is the moment of inertia, m is the mass of the sphere, and r is the radius of the sphere. Rotating unbalance is the uneven distribution of mass around an axis of rotation. A sketch of the rod, volume element, and axis is shown in Figure \ (\PageIndex {2}\). In integral form the moment of inertia is . m = kg . A larger rotational inertia will result in a slower oscillation period and a smaller rotational inertia will result in a faster oscillation period. A disk has most of its mass closer to the axis of rotation, resulting in a lower rotational inertia. – Radius (r) is the distance from the axis of rotation in meters. Step 3: Integrate. This is why objects with larger rotational inertia, such as a spinning top, tend to have a more stable and consistent motion. 3. Aug 11, 2021 · Moment of Inertia. Moments of Inertia. Jul 20, 2022 · Using our kinematics result that the tangential acceleration is aθ, i = riαz, where αz is the z -component of angular acceleration, we have that. Moment of Inertia Formula. Step 2: Find the integral limits with respect to the axis of rotation. In the previous chapter we found that if a great many forces are acting on a complicated mass of particles, whether the particles comprise a rigid or a nonrigid body, or a cloud of stars, or anything else, and we find the sum of all the forces (that is, of course, the external forces, because the internal forces balance out), then if we consider the body as a whole, and say it has a total mass Apr 30, 2007 · The formula for calculating the rotational inertia of a rectangular prism is I = (1/12) * m * (h^2 + l^2), where I is the moment of inertia, m is the mass of the prism, h is the height of the prism, and l is the length of the prism. Understanding rotational inertia is essential in various fields, from engineering to astrophysics, where rotating objects abound. Aug 13, 2020 · Calculating the Rotational Inertia. Similarly, the greater the moment of inertia of a Determining the Radius of Gyration. Relation Between Torque And Moment Of Inertia. We defined the moment of inertia I of an object to be. 1: No. Problem 4: Calculate the rotational inertia of a Jun 17, 2019 · The moment of inertia of one blade is that of a thin rod rotated about its end, listed in Figure 4. Angular Acceleration: Angular acceleration is the acceleration of an object due to rotational motion 1: No. 2 6. The linear power when the force is a constant is P = . Therefore the rotational inertia of the object will be 980 kg m². This result is also a standard formula in physics. 2. I = 1 12 × × 2. The inertia of an object does not depend on where the mass is within the body, only on how much mass there is. m/s^2 (N) times meter = kg. and the moment of inertia of a thin spherical shell is. 1. moment of inertia, in physics, quantitative measure of the rotational inertia of a body—i. Calculate the rotational Sep 12, 2022 · Power for rotational motion is equally as important as power in linear motion and can be derived in a similar way as in linear motion when the force is a constant. What is the unit of measurement for rotational inertia? The unit for rotational inertia is kilogram 5. Thus, I = 4Ml2 3 = 4 × (50. Step 1: Express dm as a function of r with the help of the density. The Parallel Axis Theorem! Jun 21, 2024 · We can calculate the magnitude of the angular velocity, or the angular frequency, as the product of the factor 2π (in radians, rad) and the frequency of the event f (in hertz, Hz): ω = 2πf. Calculate the rotational inertia for a solid cylinder or disk of radius "r" and mass "m" by the formula, inertia =1/2(m)(r)(r). I know a is acceleration, but I'm not exactly sure how that factors out. Oct 19, 2023 · Problem 3: Determine the rotational inertia of a hoop (thin-walled ring) with radius R and mass M rotating about its central axis. τ = mr2α. (b) Calculate the translational kinetic energy of the helicopter when it flies at 20. 4 simplifies and the net torque can be taken out of Oct 30, 2005 · The formula for calculating the rotational inertia of a spool is I = 0. Moment of inertia of a bar rotating around its centre calculation. τ = m r 2 α. The moment of inertia depends not only on the mass and shape of the object but also on the axis of rotation. In this problem, we will calculate the moment of inertia about an axis perpendicular to the rod that passes through the center of mass of the rod. The Torque in rotational motion is equivalent to force in linear motion. Problems & Exercises. J J is a rotational quantum number ranging from J = 0 J = 0 to J = ∞ J = ∞. The resulting angular acceleration is Mar 12, 2024 · The quantity mr2 m r 2 is called the rotational inertia or moment of inertia of a point mass m m a distance r r from the center of rotation. To fully utilize Newton's second law in rotational form, we must be able to set up and evaluate the integral that determines the rotational inertia. There are three core principles to understand when using a rotational kinetic energy calculator: 1. m/s^2, so now you know how to write Newton in basic units. Apr 19, 2015 · Rotational inertia is directly related to an object's rotational kinetic energy. (→τS, i)z = riFθ, i = Δmir2 iαz. The rotational kinetic energy is J. Rotational inertia plays the same role in rotational motion as mass plays in linear motion. 1 The student is able to describe or calculate the angular momentum and rotational inertia of a system in terms of the locations and velocities of objects that make up the system. 00m)2 3 = 1067. I (spherical shell) = kg m 2. 5. The moment of inertia of any object about an axis through its CG can be expressed by the formula: I = Mk 2 where I = moment of inertia. From this you take just the units: Newton = kg . = 980 kg m². The moment of inertia of a sphere about its central axis and a thin spherical shell are shown. Links. Dec 22, 2020 · Angular momentum (the rotational analogue for linear momentum) is defined as the product of the rotational inertia (i. or. Angular displacement (φ) rad. This is because a larger rotational inertia requires more energy to change the direction of the pendulum's swing, resulting in a Jun 25, 2019 · Moment of inertia, or Rotational Inertia, is denoted in mechanics by the letter I. The quantity mr 2 is called the rotational inertia or moment of inertia of a point mass m a distance r from the center of Mar 22, 2019 · Put simply, the rotational inertia (represented by \ (I\)) of an object is a measure of how much a spinning object will "resist" deviating from a uniform and constant angular velocity \ (\vec {ω}\). Apr 16, 2020 · A hoop has a higher rotational inertia compared to a disk of the same mass and size because its mass is concentrated at the outer edge, which is farther away from the axis of rotation. The following indicates some of the equations for different objects when calculating load inertia, for simplicity, the axis of rotation will be about the center of the object or central axis. Gear Design in 3D & Learning; Matching 3D parts for rotational inertia calculator calculation. Calculate the rotational inertia of a meter stick, with mass 0. Moment of Inertia (I) = Σ m i r i 2. It is also a crucial concept in fields such as physics, engineering, and astronomy. Tire Data Information. Number i Units. Solution: The formula for the rotational inertia of a hoop is I = MR², where M is the mass and R is the radius. That is, rF = mr2α. 3. By understanding this fundamental property, professionals across multiple disciplines can better solve problems tied to rotational dynamics and design more Apr 10, 2024 · Moment of Inertia. 3 cm. Inertia is dependent on mass and the radius or length of the object and the axis of rotation. . This last equation is the rotational analog of Newton’s second law ( F = ma ) where torque is analogous to force, angular acceleration is analogous to translational acceleration, and mr2 is analogous to mass (or inertia). Moment of Inertia: Sphere. (To be honest, this is a lie. Inertia is the tendency of objects in motion to stay in motion and objects at rest to stay at rest, unless a force causes its speed or direction to change. kg m 2. \text L = \text I×\text ω L = I×ω. GD&T. This assumes that the spool has a uniform density and is rotating around its central axis. It's the angular esistance to keep the stationary object stationary and moving object moving, mathematically derived from the sum of products obtained by multiplying the mass of each particle of matter in a given objevt by the square of its distance from the axis Moment of inertia is the property of the body due to which it resists angular acceleration, which is the sum of the products of the mass of each particle in the body with the square of its distance from the axis of rotation. 25 Calculation of the moment of inertia I for a uniform thin rod about an axis through the center of the rod. 1 cm mark. Spherical Gear (Z-axis) Elbows or. ) Number i Units. Rotational inertia, rotational inertia is the measure of an object’s resistance to change in its rotation. Compute the radius of the circular path. Students are expected to do qualitative reasoning with compound objects. Load inertia is typically referred to as "J". It is one of the fundamental principles in classical physics, and described by Isaac Newton in his first law of motion (also known as The Principle of Inertia). The axis may be internal or external and may or may not be fixed. Where I is the moment of inertia, M is the mass of the object, and R is its radius. Because \ (r\) is the distance to the axis of rotation from each piece of mass that makes up the object, the moment of inertia for any object depends on the chosen axis. Jun 17, 2019 · Figure \(\PageIndex{7}\): Calculating the moment of inertia for a child on a merry-go-round. Moment of inertia depends on both mass and its distribution relative to the axis of rotation. The rotational energy levels of a diatomic molecule in 3D space is given by the quantum mechanical solution to the rotating rigid rotor: E = J(J + 1)ℏ2 2I (1) (1) E = J ( J + 1) ℏ 2 2 I. Apr 4, 2011 · These equations are then solved to find the unknown rotational inertia of the pulley. The Moment of Inertia of a spherical shell rotating around its diameter is kg∙m 2. m^2/s^2. Step 4: Substitute the density to get the answer in terms of the mass of the body. Calculate the rotational energy levels for l = 1 and l = 2 using the relation El = l (l+1)hbar2/ (2/I). From Equation (17. 8. This last equation is the rotational analog of Newton’s second law ( F = ma F = m a ), where torque is analogous to force, angular acceleration is analogous to translational acceleration, and mr2 m r 2 is analogous to mass (or inertia). Mar 27, 2008 · For rotational energies, the total energy is due to rotational kinetic energy. where. What is the formula for calculating rotational inertia? The rotational inertia is a property of any object which rotates. where, m = Sum of the product of the mass. Here are some common examples: 1. bottom line - unit of the Moment of inertia: I=m. There are several ways to calculate the moment of inertia of a rotating object. Because of this, any symmetry axis of the shape, is also a principal axis. The rotational inertia is various with the object depending on the rotational axis. m: Mass of the Object, in Kg. Nov 13, 2009 · The moment of inertia is a physical property of an object that describes its resistance to rotational motion. Using the formula we have, I = mr 2 = 20 × 4 × 4 = 320 kg m 2. (D) calculate the effect of forces on objects, including the law of inertia, the relationship between force and acceleration, and the nature of force pairs between objects. Torque depends on three factors: force magnitude, force direction, and point of application. You’ll undoubtedly be familiar with the law of conservation of linear momentum, and angular momentum is also How large does the lump need to be to just rotate the circle? Describe how you can change the moment of inertia of the circle. Consider a thin uniform rod of length \ (L\) and mass \ (m\). Dec 22, 2021 · Yes. the moment of inertia of a solid sphere is. We are given the mass and distance to the axis of rotation of the child as well as the mass and radius of the merry-go-round. For the vast majority of common shapes, and many quite uncommon shapes, these integrals have already been evaluated. Sharing is Caring. I I is the moment of inertia and is Rotational inertia Formula is given by, I = mr². The angular momentum calculator uses both these formulas to calculate the angular momentum of an object for your ease. Question 2. May 1, 2023 · Published: May 1, 2023. Rotational Inertia. It is commonly denoted by the symbol I and is calculated by summing the mass of each particle in the object multiplied by the square of its distance from the axis of rotation. Principal axes. 4. When the rotational inertia of an object is constant, the angular acceleration is proportional to torque. This problem involves the calculation of a moment of inertia. If the net torque is constant over the angular displacement, Equation 10. The formula for calculating rotational inertia is I = mr^2, where I is the moment of inertia, m is the mass of the object, and r is the distance from the axis of rotation to the object's mass. This last equation is the rotational analog of Newton’s second law (F=ma), where torque is analogous to force, angular acceleration is analogous to translational acceleration, and mr 2 is analogous to mass (or inertia). m^2. In Figure, block 1 has mass m1 = 430 g, block 2 has mass m2 = 540 g, and the pulley is on a frictionless horizontal axle and has radius R = 5. 4. This physics video tutorial provides a basic introduction into angular momentum. The moment of inertia calculates the rotational inertia of an object rotating around a given axis. Calculate the rotational inertia for an object of mass 20 kg and radius 4 m. Sep 12, 2022 · In this subsection, we show how to calculate the moment of inertia for several standard types of objects, as well as how to use known moments of inertia to find the moment of inertia for a shifted axis or for a compound object. Fθ, i = Δmiriαz. To calculate rotational inertia for simple shapes like cylinders, spheres, rods, and disks, we can use established formulas. ) We find that with rotation, it's not the amount of mass that matters most; it's the way the mass is distributed Mar 4, 2024 · The Rotational Inertia Calculator plays a crucial role in quantifying this resistance, providing a numerical representation of how mass is distributed relative to the axis of rotation. What is the Torque on an Object from its Inertia. 2. The purpose of this experiment is to find the rotational inertia of a ring and a disk experimentally and to verify that these values correspond to the calculated theoretical values. ra aw ku ek je ds ym ro pe en