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Problem Statement Suppose we have a uniform rod of mass M and length L that can pivot about one end. The other end is attached to a horizontal spring with constant k that is affixed to a wall. The spring is neither stretched nor compressed when the rod hangs straight down. Assume that the rod's angle from the vertical is always small and that the spring does not bend or bow. L k M A. Draw a free body diagram for the pendulum and be sure to explicitly label any angles, forces, locations of forces, center of mass, etc. For full credit, students will follow the procedure for drawing a free-body diagram laid out in the document "A guide to free-body diagrams." B. Using the forces identified in your free body diagram, write down Newton's law of torque for the physical pendulum. C. Derive an equation of motion for the pendulum. See Model 15.1 "simple harmonic motion" from your textbook (Ch. 15.6, pg. 406). Hint: You may need to look up some trigonometric identities and recall the small angle approximation. D. Deduce a symbolic expression for the angular frequency $\omega$ of the pendulum using your equation of motion. Again, see Model 15.1 and Ch. 15.6 of the textbook. E. Assess the validity of the expression you found for the angular frequency $\omega$ by examining at least 3 limiting cases in terms of the variables k, g, L, and m. To do so, first comment in one sentence what you expect to happen if k, g, L, or m increase/decrease. Then show that your symbolic expression for the angular frequency $\omega$ backs up your intuition when this change is made.

          Problem Statement
Suppose we have a uniform rod of mass M and length L that can pivot about one end. The other
end is attached to a horizontal spring with constant k that is affixed to a wall. The spring is neither
stretched nor compressed when the rod hangs straight down. Assume that the rod's angle from the
vertical is always small and that the spring does not bend or bow.
L
k
M
A. Draw a free body diagram for the pendulum and be sure to explicitly label any angles, forces,
locations of forces, center of mass, etc. For full credit, students will follow the procedure for
drawing a free-body diagram laid out in the document "A guide to free-body diagrams."
B. Using the forces identified in your free body diagram, write down Newton's law of torque
for the physical pendulum.
C. Derive an equation of motion for the pendulum. See Model 15.1 "simple harmonic motion"
from your textbook (Ch. 15.6, pg. 406). Hint: You may need to look up some trigonometric
identities and recall the small angle approximation.
D. Deduce a symbolic expression for the angular frequency $\omega$ of the pendulum using your
equation of motion. Again, see Model 15.1 and Ch. 15.6 of the textbook.
E. Assess the validity of the expression you found for the angular frequency $\omega$ by examining at
least 3 limiting cases in terms of the variables k, g, L, and m. To do so, first comment in one
sentence what you expect to happen if k, g, L, or m increase/decrease. Then show that your
symbolic expression for the angular frequency $\omega$ backs up your intuition when this change is
made.

Problem Statement
Suppose we have a uniform rod of mass M and length L that can pivot about one end. The other
end is attached to a horizontal spring with constant k that is affixed to a wall. The spring is neither
stretched nor compressed when the rod hangs straight down. Assume that the rod's angle from the
vertical is always small and that the spring does not bend or bow.
L
k
M
A. Draw a free body diagram for the pendulum and be sure to explicitly label any angles, forces,
locations of forces, center of mass, etc. For full credit, students will follow the procedure for
drawing a free-body diagram laid out in the document "A guide to free-body diagrams."
B. Using the forces identified in your free body diagram, write down Newton's law of torque
for the physical pendulum.
C. Derive an equation of motion for the pendulum. See Model 15.1 "simple harmonic motion"
from your textbook (Ch. 15.6, pg. 406). Hint: You may need to look up some trigonometric
identities and recall the small angle approximation.
D. Deduce a symbolic expression for the angular frequency ω of the pendulum using your
equation of motion. Again, see Model 15.1 and Ch. 15.6 of the textbook.
E. Assess the validity of the expression you found for the angular frequency ω by examining at
least 3 limiting cases in terms of the variables k, g, L, and m. To do so, first comment in one
sentence what you expect to happen if k, g, L, or m increase/decrease. Then show that your
symbolic expression for the angular frequency ω backs up your intuition when this change is
made.

Added by James S.

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