A mechanical system consists of a ("long") rigid rod of length d and mass M, and another ("short") rigid rod of length I and mass m. The rods are attached at a fixed 90° angle such that the endpoint of the long rod touches the middle of the short rod; see the figure below. The mass per unit length of the rods is constant. The system can move in a vertical plane (x, y) with the endpoint O of the long rod fixed, so that the system is free to oscillate in the vertical plane about the point O. Gravity acts along the vertical y direction. a) How many degrees of freedom does the system have? [Expect to use a line to answer this question.] X b) Calculate the moment of inertia of the system constituted by the rods about an axis orthog- onal to the plane (x,y) and passing through O. Assume that the widths of the rods are negligible, so that you can effectively treat them as one-dimensional objects. [Expect to use about half a page to answer this question.] c) Determine the distance of the centre of mass of the system from the point O, and write down the gravitational potential of the system. [Expect to use about half a page to answer this question.] d) Write down the Lagrangian of the system and the Euler-Lagrange equations. [Expect to use about half a page to answer this question.] e) Determine the frequency of small oscillations w about the equilibrium position 0 = 0 (the angle is defined in the figure). In what limit does the system reduce to a simple planar pendulum? (A simple pendulum is a point particle moving at a distance d from point O.) Check that w in this limit reduces to the well-known formula for a simple pendulum. [Expect to use about half a page to answer this question.]
A mechanical system consists of a ("long") rigid rod of length d and mass M, and another ("short") rigid rod of length I and mass m. The rods are attached at a fixed 90° angle such that the endpoint of the long rod touches the middle of the short rod; see the figure below. The mass per unit length of the rods is constant. The system can move in a vertical plane (x, y) with the endpoint O of the long rod fixed, so that the system is free to oscillate in the vertical plane about the point O. Gravity acts along the vertical y direction. a) How many degrees of freedom does the system have? [Expect to use a line to answer this question.] X b) Calculate the moment of inertia of the system constituted by the rods about an axis orthog- onal to the plane (x,y) and passing through O. Assume that the widths of the rods are negligible, so that you can effectively treat them as one-dimensional objects. [Expect to use about half a page to answer this question.] c) Determine the distance of the centre of mass of the system from the point O, and write down the gravitational potential of the system. [Expect to use about half a page to answer this question.] d) Write down the Lagrangian of the system and the Euler-Lagrange equations. [Expect to use about half a page to answer this question.] e) Determine the frequency of small oscillations w about the equilibrium position 0 = 0 (the angle is defined in the figure). In what limit does the system reduce to a simple planar pendulum? (A simple pendulum is a point particle moving at a distance d from point O.) Check that w in this limit reduces to the well-known formula for a simple pendulum. [Expect to use about half a page to answer 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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![A mechanical system consists of a ("long") rigid rod of length d and mass M, and another ("short")
rigid rod of length I and mass m. The rods are attached at a fixed 90° angle such that the endpoint
of the long rod touches the middle of the short rod; see the figure below. The mass per unit length
of the rods is constant. The system can move in a vertical plane (x, y) with the endpoint O of
the long rod fixed, so that the system is free to oscillate in the vertical plane about the point O.
Gravity acts along the vertical y direction.
0
d
a)
How many degrees of freedom does the system have?
[Expect to use a line to answer this question.]
â
b) Calculate the moment of inertia of the system constituted by the rods about an axis orthog-
onal to the plane (x, y) and passing through O. Assume that the widths of the rods are
negligible, so that you can effectively treat them as one-dimensional objects.
[Expect to use about half a page to answer this question.]
c) Determine the distance of the centre of mass of the system from the point O, and write
down the gravitational potential of the system.
[Expect to use about half a page to answer this question.]
d) Write down the Lagrangian of the system and the Euler-Lagrange equations.
[Expect to use about half a page to answer this question.]
e) Determine the frequency of small oscillations w about the equilibrium position 0 = 0 (the
angle is defined in the figure). In what limit does the system reduce to a simple planar
pendulum? (A simple pendulum is a point particle moving at a distance d from point O.)
Check that w in this limit reduces to the well-known formula for a simple pendulum.
[Expect to use about half a page to answer this question.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4b7af157-3088-4e07-9322-eb6941ca83f4%2Fc0dcda55-409c-421e-8001-b1f30a836973%2Fqu1yvki_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A mechanical system consists of a ("long") rigid rod of length d and mass M, and another ("short")
rigid rod of length I and mass m. The rods are attached at a fixed 90° angle such that the endpoint
of the long rod touches the middle of the short rod; see the figure below. The mass per unit length
of the rods is constant. The system can move in a vertical plane (x, y) with the endpoint O of
the long rod fixed, so that the system is free to oscillate in the vertical plane about the point O.
Gravity acts along the vertical y direction.
0
d
a)
How many degrees of freedom does the system have?
[Expect to use a line to answer this question.]
â
b) Calculate the moment of inertia of the system constituted by the rods about an axis orthog-
onal to the plane (x, y) and passing through O. Assume that the widths of the rods are
negligible, so that you can effectively treat them as one-dimensional objects.
[Expect to use about half a page to answer this question.]
c) Determine the distance of the centre of mass of the system from the point O, and write
down the gravitational potential of the system.
[Expect to use about half a page to answer this question.]
d) Write down the Lagrangian of the system and the Euler-Lagrange equations.
[Expect to use about half a page to answer this question.]
e) Determine the frequency of small oscillations w about the equilibrium position 0 = 0 (the
angle is defined in the figure). In what limit does the system reduce to a simple planar
pendulum? (A simple pendulum is a point particle moving at a distance d from point O.)
Check that w in this limit reduces to the well-known formula for a simple pendulum.
[Expect to use about half a page to answer this question.]
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