Engineering Mechanics: Statics & Dynamics (14th Edition)

Published by Pearson
ISBN 10: 0133915425
ISBN 13: 978-0-13391-542-6

Chapter 19 - Planar Kinetics of a Rigid Body: Impulse and Momentum - Section 19.2 - Principle of Impulse and Momentum - Problems - Page 537: 18


$\omega=3.90 rad/s$

Work Step by Step

We can determine the required angular velocity as follows: First, we apply the impulse momentum principle in y-direction $mv_{Gy1}+\Sigma \int_{t_1}^{t_2} F_y dt=mv_{Gy2}$ $\implies mv_{Gy1}+Isin60=mv_{Gy2}$ $\implies 0+8sin60=4v_{Gy2}$ This simplifies to: $v_{Gy2}=\sqrt{3}m/s$ Now we apply the impulse momentum principle in x-direction $mv_{Gx1}+\Sigma \int_{t_1}^{t_2} F_x dt=mv_{Gx2}$ $\implies mv_{Gx1}+Icos60=mv_{Gx2}$ This simplifies to: $v_{Gx2}=1m/s$ Thus, $v_G=\sqrt{v_{Gx}^2+v_{Gy}^2}=\sqrt{(1)^2+\sqrt{3}^2}=2m/s$ Now, we can calculate the angular velocity $I_A\omega_1+\Sigma\int_{t_1}^{t_2} M_Gdt=I_A\omega_2$ [eq(1)] As $I_A=\frac{1}{12}mL^2=\frac{1}{12}(4)(2)^2=\frac{4}{3}Kg\cdot m^2$ and $\omega_1=0$ Similarly $\Sigma\int_{t_1}^{t_2} M_Gdt=Isin60\times 0.75=3\sqrt {3} N\cdot s$ We plug in the known values in eq(1) to obtain: $0+3\sqrt{3}=\frac{4}{3}\omega$ This simplifies to: $\omega=3.90 rad/s$
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