Engineering Mechanics: Dynamics (14th Edition)
14th Edition
ISBN: 9780133915389
Author: Russell C. Hibbeler
Publisher: PEARSON
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Chapter 21.3, Problem 40P
To determine
Derive the scalar form of the rotational equation of motion about the x axis.
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Chapter 21 Solutions
Engineering Mechanics: Dynamics (14th Edition)
Ch. 21.1 - Show that the sum of the moments of inertia of a...Ch. 21.1 - Determine the moment of inertia of the cone with...Ch. 21.1 - Determine moment of inertia Iy of the solid formed...Ch. 21.1 - Determine the moments of inertia Ix and Iy of the...Ch. 21.1 - The density of the material is . Express the...Ch. 21.1 - Prob. 6PCh. 21.1 - Prob. 7PCh. 21.1 - Prob. 8PCh. 21.1 - The weight of the cone is 15 lb, the height is h =...Ch. 21.1 - The density of the material is .
Ch. 21.1 - Prob. 11PCh. 21.1 - Determine the moment of inertia Ixx of the...Ch. 21.1 - Determine the product of inertia Iyz of the...Ch. 21.1 - Prob. 14PCh. 21.1 - Prob. 15PCh. 21.1 - Determine the moment of inertia of the rod about...Ch. 21.1 - Prob. 17PCh. 21.1 - Prob. 18PCh. 21.1 - Prob. 19PCh. 21.1 - Prob. 20PCh. 21.1 - Prob. 21PCh. 21.3 - If a body contains no planes of symmetry, the...Ch. 21.3 - Prob. 23PCh. 21.3 - Prob. 24PCh. 21.3 - The large gear has a mass of 5 kg and a radius of...Ch. 21.3 - Prob. 26PCh. 21.3 - Prob. 27PCh. 21.3 - Prob. 28PCh. 21.3 - Prob. 29PCh. 21.3 - Prob. 30PCh. 21.3 - Prob. 31PCh. 21.3 - Prob. 32PCh. 21.3 - The 20-kg sphere rotates about the axle with a...Ch. 21.3 - The 200-kg satellite has its center of mass at...Ch. 21.3 - Prob. 35PCh. 21.3 - Prob. 36PCh. 21.3 - Prob. 37PCh. 21.3 - Prob. 38PCh. 21.3 - Prob. 39PCh. 21.3 - Prob. 40PCh. 21.4 - Derive the scalar form of the rotational equation...Ch. 21.4 - Prob. 42PCh. 21.4 - Prob. 43PCh. 21.4 - Prob. 44PCh. 21.4 - The disk has a weight of 15 lb. Neglect the weight...Ch. 21.4 - Prob. 46PCh. 21.4 - Prob. 47PCh. 21.4 - Prob. 48PCh. 21.4 - Prob. 49PCh. 21.4 - Prob. 50PCh. 21.4 - Prob. 51PCh. 21.4 - The 5-kg circular disk is mounted off center on a...Ch. 21.4 - Prob. 53PCh. 21.4 - Prob. 54PCh. 21.4 - Prob. 55PCh. 21.4 - The 4-kg slender rod AB is pinned at A and held at...Ch. 21.4 - Prob. 57PCh. 21.4 - Prob. 58PCh. 21.4 - Prob. 59PCh. 21.4 - Show that the angular velocity of a body, in terms...Ch. 21.4 - Prob. 61PCh. 21.6 - The gyroscope consists of a uniform 450-g disk D...Ch. 21.6 - Prob. 63PCh. 21.6 - Prob. 64PCh. 21.6 - Prob. 65PCh. 21.6 - When viewed from the front of the airplane, the...Ch. 21.6 - Prob. 67PCh. 21.6 - Prob. 68PCh. 21.6 - Prob. 69PCh. 21.6 - Prob. 70PCh. 21.6 - Prob. 71PCh. 21.6 - Prob. 72PCh. 21.6 - Prob. 73PCh. 21.6 - Prob. 74PCh. 21.6 - Prob. 75PCh. 21.6 - Prob. 76PCh. 21.6 - Prob. 77PCh. 21.6 - Prob. 78P
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- Consider a satellite moving in a torque-free environment. The inertial frame and satellite body-fixed frame are represented by N-frame and B-frame, where {^₁, ŵ2, ñ3} and {b₁, 62, 63} are right-handed vector bases fixed in N-frame and B-frame, respectively. {1;} is aligned with the principal axis of inertia, where ☎₁ and 3 are associated with the minimum and maximum moments of inertia, respectively. Denote the satellite angular velocity by w = wib; and the inertia tensor about the CoM by Ic. Suppose that I at time t = 0 is given by: Ic(t = 0) = 2ñ₁ñ₁ +2.25 ñ¿Ñ₂ +2.25 ñ³Ñ3 – 0.75 ñ¿ñ3 −0.75 3₂ (dimensionless) Express the satellite inertia tensor about the CoM in B-frame. Determine [BN] at time t = 0, i.c., the direction cosine matrix (DCM) that represents the orientation of B-frame relative to N-frame.arrow_forwardGiven that P = 50N, and the rod has mass = 0.370 kg with centroidal mass moment of inertia l = 37/19200 kg-m²:a. Which of the equations given in the second image can be used to solve for the angular acceleration of rod BD?b. What is the angular acceleration of rod BD?arrow_forwardThe work done by spring is The work done by weight is Polar moment of inertia about B is angular velocity at position 2arrow_forward
- 2. Explain in your own words the parallel-axis theorem and how it is important in the analysis of rotational dynamics.arrow_forwardA mass m is hanging from a cord attached to the circular homogeneous disk (at point A) of mass M and radius R, as shown in Figure 1. The disk is restrained from rotating by a spring attached at point B at radius r from the center (point O). There is no friction at the pivot. M Disk Mass Moment of Inertia: Figure 1arrow_forwardConsider a satellite moving in a torque-free environment. The inertial frame and satellite body-fixed frame are represented by N-frame and B-frame, where {^₁, ñ2, ñ3} and {b₁, 62, 63} are right-handed vector bases fixed in N-frame and B-frame, respectively. {b} is aligned with the principal axis of inertia, where 6₁ and 63 are associated with the minimum and maximum moments of inertia, respectively. Denote the satellite angular velocity by w = wibį and the inertia tensor about the CoM by Ie. Suppose that I at time t = 0 is given by: Ic(t = 0) = 2ñ₁ñ₁ +2.25 ñ₂n₂ +2.25 ñ3ñ3 -0.75 n₂n3 -0.75 n3n2 (dimensionless) (a): First, let us characterize the inertia property of the satellite from the given information. (a.1): Express the satellite inertia tensor about the CoM in B-frame. Sketch the inertia ellipsoid of this body with B-frame.arrow_forward
- 6) For the system shown, assume x-y-z is the inertial frame. If the bar first twists around A about the z-axis by a-70°, then the bar BD spins by 0-30° (shown in the picture). Define the transformation matrix to the Bar frame (B) from the Inertia frame (I) and find the location of C with respect to point O expressed in the inertial frame. (Assume a=0 as shown in the picture). 2 ft r=1 ft. B @=30⁰ 1.5 ftarrow_forwardFigure shows a mechanical system. The connecting link has moment of inertia J about its pivot point, and rotation angle is positive clockwise. Position of mass m is positive to the right. Both the angular and translational displacements are measured from the equilibrium position where all springs are undeflected. Derive the mathematical model of this system assuming small rotation angle 8. Link, moment. of inertia J L k₁ www m O k₂ wwwarrow_forwardA dancer is spinning at 72 rpm about an axis through her center with her arms outstretched, as shown in (Figure 1). From biomedical measurements, the typical distribution of mass in a human body is as follows: Part A Head: 7.0% Arms: 13% (for both) Trunk and legs: 80.0% Calculate moment of inertia about the dancer's spin axis. Express your answer with the appropriate units. Suppose the mass of the dancer is 64.5 kg, the diameter of her head is 16 cm, the width of her body is 24 length of her arms is 60 cm. cm, and the HÀ ? I = Value Units Submit Request Answer Figure 1 of 1 > Part B Calculate dancer's rotational kinetic energy. Express your answer with the appropriate units. ? K = Value Units Submit Request Answerarrow_forward
- A wheel in fixed axis rotation spins at 300 rpm and decelerates at α = 3 rad/s2. How long does it take for the wheel to come to a complete stop?arrow_forwardA car is moving on a curved horizontal road of radius 100 m with a speed of 20 m/s. The rotating masses of the engine have an angular speed of 100 rad/s in clockwise direction when viewed from the front of the car The combined moment of inertia of the rotating masses is 10 kg-m².what is the magnitude of the gyroscopic moment in (N-m)?arrow_forward3. For the diagram below, derive for the angular displacements of the rotating discs. GJ₁ L₁ M₁(t) 0₁ (1) >|< GJ₂ L2- M₂(t) 0₂(1)arrow_forward
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