Assignment-1-Solutions

Assignment-1-Solutions - Chapter 2 Kinematics Question 21 A...

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Chapter 2 Kinematics Question 2–1 Abug B crawls radially outward at constant speed v 0 from the center of a ro- tating disk as shown in Fig. P2-1. Knowing that the disk rotates about its cen- ter O with constant absolute angular velocity Ω relative to the ground (where ± Ω ±= Ω ), determine the velocity and acceleration of the bug as viewed by an observer ±xed to the ground. r v 0 B O Ω Figure P2-1 Solution to Question 2–1 For this problem it is convenient to choose a ±xed reference frame F and a non- inertial reference frame A that is ±xed in the disk. Corresponding to reference frame F we choose the following coordinate system: Origin at Point O E x = Along OB at Time t = 0 E z = Out of Page E y = E z × E x
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2 Chapter 2. Kinematics Corresponding to the reference frame A that is Fxed in the disk, we choose the following coordinate system Origin at Point O e x = Along OB e z = Out of Page ( = E z ) e y = e z × e x The position of the bug is then resolved in the basis { e x , e y , e z } as r = r e x (2.1) Now, since the platform rotates about the e z -direction relative to the ground, the angular velocity of reference frame A in reference frame F is given as F ω A = Ω e z (2.2) The velocity is found by applying the basic kinematic equation. This gives F v = F d r dt = A d r dt + F ω A × r (2.3) Now we have A d r dt = ˙ r e x = v 0 e x (2.4) F ω A × r = Ω e z × r e x = Ω r e y (2.5) Adding Eqs. (2.4) and (2.5), we obtain the velocity of the bug in reference frame F as F v = v 0 e x + Ω r e y (2.6) The acceleration is found by applying the basic kinematic equation to F v .Th is gives F a = F d dt ± F v ² = A d dt ± F v ² + F ω A × F v (2.7) Using F v from Eq. (2.6) and noting that v 0 and Ω are constant, we have that A d dt ± F v ² = Ω ˙ r e y = Ω v 0 e y F ω A × F v = Ω e z × [v 0 e x + r Ω e y ] =- Ω 2 r e x + Ω v 0 e y (2.8) Therefore, the acceleration in reference frame F is given as F a Ω 2 r e x + 2 Ω v 0 e y (2.9)
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3 Question 2–2 A particle, denoted by P , slides on a circular table as shown in Fig. P2-2. The position of the particle is known in terms of the radius r measured from the center of the table at point O and the angle θ where θ is measured relative to the direction of OQ where Q is a point on the circumference of the table. Knowing that the table rotates with constant angular rate Ω , determine the velocity and acceleration of the particle as viewed by an observer in a ±xed reference frame. r O P Q θ Ω Figure P2-2 Solution to Question 2–2 For this problem it is convenient to de±ne a ±xed inertial reference frame F and two non-inertial reference frames A and B . The ±rst non-inertial reference frame A is ±xed to the disk while the second non-inertial reference frame B is ±xed to the direction of OP . Corresponding to the ±xed inertial reference frame F , we choose the following coordinate system: Origin at point O E x = Along Ox at t = 0 E z = Out of Page E y = E z × E x Corresponding to non-inertial reference frame A , we choose the following co- ordinate system: Origin at point O e x = Along e z = Out of Page ( = E z ) e y = e z × e x
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4 Chapter 2. Kinematics
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Assignment-1-Solutions - Chapter 2 Kinematics Question 21 A...

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