A particle P of mass m = 1 kg arrives at point with a horizontal velocity vo and starts rolling downa circle with radius R = 1 m. The particle is attached to the point O by means of a linear spring ofstiffness k = 50 N/m that is unstretched when the particle is at O. Assume that during the motion ofP along the circle, the spring follows the circle. Let N = {ĥ, ny} be a fixed reference frame asshown in the figure and let B = {b, by} be a reference frame attached to P so that b is tangentto the circular path and by is perpendicular to the circular path. For convenience, take g = 10 m/s²in this question.ny1/nxIP a)Determine the Direction Cosine Matrices [C]B and BCNb)N₂Determine for p in the frame N the position vector NTP, velocity vector NVP, and accelerationvector NaPc)Draw a Free Body Diagram of the particle, clearly indicating all forces acting on the particle, anduse the Free Body Diagram to determine the equation(s) of motion of the particle.d)What relationship between 0 and needs to be satisfied at the point where the particle leavesthe circle?e)For each value of between 0 and determine the value of vo for which the particle leavesthe circular path at 0 = 00.

A particle P of mass m = 1 kg arrives at point O with a horizontal velocity vo and starts rolling down a circle with radius R = 1 m. The particle is attached to the point O by means of a linear spring of stiffness k = 50 N/m that is unstretched when the particle is at O. Assume that during the motion of P along the circle, the spring follows the circle. Let N = {n, ny} be a fixed reference frame as shown in the figure and let B = {ĥ, by } be a reference frame attached to P so that ĥ is tangent to the circular path and by is perpendicular to the circular path. For convenience, take g = 10 m/s² in this question.

A particle P of mass m = 1 kg arrives at point O with a horizontal velocity vo and starts rolling down a circle with radius R = 1 m. The particle is attached to the point O by means of a linear spring of stiffness k = 50 N/m that is unstretched when the particle is at O. Assume that during the motion of P along the circle, the spring follows the circle. Let N = {n, ny} be a fixed reference frame as shown in the figure and let B = {ĥ, by } be a reference frame attached to P so that ĥ is tangent to the circular path and by is perpendicular to the circular path. For convenience, take g = 10 m/s² in this question.

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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Conservation of Energy

In physics, the rule of the conservation of energy states that energy doesn’t disappear from the atmosphere; instead, it changes its form from one type to another. Energy is one of the most important physical quantities in nature, and it cannot be created or destroyed — only transformed.

Question

A particle P of mass m = 1 kg arrives at point with a horizontal velocity vo and starts rolling down
a circle with radius R = 1 m. The particle is attached to the point O by means of a linear spring of
stiffness k = 50 N/m that is unstretched when the particle is at O. Assume that during the motion of
P along the circle, the spring follows the circle. Let N = {ĥ, ny} be a fixed reference frame as
shown in the figure and let B = {b, by} be a reference frame attached to P so that b is tangent
to the circular path and by is perpendicular to the circular path. For convenience, take g = 10 m/s²
in this question.
ny
1/
nx
IP

a)
Determine the Direction Cosine Matrices [C]B and BCN
b)
N₂
Determine for p in the frame N the position vector NTP, velocity vector NVP, and acceleration
vector NaP
c)
Draw a Free Body Diagram of the particle, clearly indicating all forces acting on the particle, and
use the Free Body Diagram to determine the equation(s) of motion of the particle.
d)
What relationship between 0 and needs to be satisfied at the point where the particle leaves
the circle?
e)
For each value of between 0 and determine the value of vo for which the particle leaves
the circular path at 0 = 00.

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