1. The small particle of mass m and its restraining cord are spinning with an angular velocity w on the horizontal surface of a smooth disk as shown in Fig. 1. The input force F,(t) applied to the cord depends on time t. As a result, the angular velocity w and the radial position r of the particle are not constant. (a) Draw a free-body diagram of the particle and shows that the angular momentum is conserved. Therefore, ho mr2 (1) where is the angular position of the particle, the dot is the time derivative, and ho is the initial angular momentum of the particle. (b) Apply Newton's second law in polar coordinates to derive the equation of motion. Sim- plify the equation in the radial direction through use of (1) to obtain h₂ m²+3 Fx(t) m (2) (c) When F,(t) = F, a constant force, the particle will undergo a circular motion. Therefore, r(t) = and w(t) = are both constant. Determine 7 and w. (d) When F,(t) undergoes a small change from F, e.g., F,(t) = F + AF(t) the radial position of the particle will deviate from the circular orbit accordingly, i.e., r(t)=r+n(t) (3) (4) Substitute (3) and (4) into (2) to linearize the equation. Show that the linearized equation takes the form of ij(t) + m²-47(t) 3 AF(t) m (5) Also, specify the initial conditions 7(0) and (0). Hint: Show that the binomial expansion of r 3 leads to 1 3 = 73(t) -+ (1)+... (6) (e) Is the linearized system a first-order system or a second-order system? If it is a first- order system, determine the time constant. If it is a second-order system, determine the natural frequency and viscous damping factor. (f) If the force increment AF is constant, determine r(t) from (4) and (5). Does the response r(t) oscillate or decay? Plot r(t) with respect to time t.
1. The small particle of mass m and its restraining cord are spinning with an angular velocity w on the horizontal surface of a smooth disk as shown in Fig. 1. The input force F,(t) applied to the cord depends on time t. As a result, the angular velocity w and the radial position r of the particle are not constant. (a) Draw a free-body diagram of the particle and shows that the angular momentum is conserved. Therefore, ho mr2 (1) where is the angular position of the particle, the dot is the time derivative, and ho is the initial angular momentum of the particle. (b) Apply Newton's second law in polar coordinates to derive the equation of motion. Sim- plify the equation in the radial direction through use of (1) to obtain h₂ m²+3 Fx(t) m (2) (c) When F,(t) = F, a constant force, the particle will undergo a circular motion. Therefore, r(t) = and w(t) = are both constant. Determine 7 and w. (d) When F,(t) undergoes a small change from F, e.g., F,(t) = F + AF(t) the radial position of the particle will deviate from the circular orbit accordingly, i.e., r(t)=r+n(t) (3) (4) Substitute (3) and (4) into (2) to linearize the equation. Show that the linearized equation takes the form of ij(t) + m²-47(t) 3 AF(t) m (5) Also, specify the initial conditions 7(0) and (0). Hint: Show that the binomial expansion of r 3 leads to 1 3 = 73(t) -+ (1)+... (6) (e) Is the linearized system a first-order system or a second-order system? If it is a first- order system, determine the time constant. If it is a second-order system, determine the natural frequency and viscous damping factor. (f) If the force increment AF is constant, determine r(t) from (4) and (5). Does the response r(t) oscillate or decay? Plot r(t) with respect to time t.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Transcribed Image Text:1. The small particle of mass m and its restraining cord are spinning with an angular velocity w
on the horizontal surface of a smooth disk as shown in Fig. 1. The input force F,(t) applied
to the cord depends on time t. As a result, the angular velocity w and the radial position r
of the particle are not constant.
(a) Draw a free-body diagram of the particle and shows that the angular momentum is
conserved. Therefore,
ho
mr2
(1)
where is the angular position of the particle, the dot is the time derivative, and ho is
the initial angular momentum of the particle.
(b) Apply Newton's second law in polar coordinates to derive the equation of motion. Sim-
plify the equation in the radial direction through use of (1) to obtain
h₂
m²+3
Fx(t)
m
(2)
(c) When F,(t) = F, a constant force, the particle will undergo a circular motion. Therefore,
r(t) = and w(t) = are both constant. Determine 7 and w.
(d) When F,(t) undergoes a small change from F, e.g.,
F,(t) = F + AF(t)
the radial position of the particle will deviate from the circular orbit accordingly, i.e.,
r(t)=r+n(t)
(3)
(4)
Substitute (3) and (4) into (2) to linearize the equation. Show that the linearized
equation takes the form of
ij(t) +
m²-47(t)
3
AF(t)
m
(5)
Also, specify the initial conditions 7(0) and (0). Hint: Show that the binomial expansion
of r 3 leads to
1 3
=
73(t)
-+
(1)+...
(6)
(e) Is the linearized system a first-order system or a second-order system? If it is a first-
order system, determine the time constant. If it is a second-order system, determine the
natural frequency and viscous damping factor.
(f) If the force increment AF is constant, determine r(t) from (4) and (5). Does the response
r(t) oscillate or decay? Plot r(t) with respect to time t.
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