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Equation of motion of the rod in terms of
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Answer to Problem 3.34P
Explanation of Solution
Given:
Wheel radius, R = 0.05m
Mass of rod, m = 20kg
Length of rod, L = 1.4m
Mass of the wheel is negligible and hence the inertia is also negligible.
The wheel does not slip.
Concept used:
The motion of this object is defined by its translational motion in the plane and its rotational motion about an axis perpendicular to the plane. Two force equations describe the translational motion, and a moment equation is needed to describe the rotational motion.
For an objects’ planar motion which rotates only about an axis perpendicular to the plane, the equation of motion can be written down using Newton’s Second Law.
Equation of Motion:
Where
Using Newton’s laws for plane motion,
Where,
Derivation of Equation of motion:
Free body diagram of the rod:
N is the reaction force on the rod.
G is the mass center of the rod.
The displacement from the mass center in the x -direction is measured with respect to the reference axis,
The displacement from the mass center in the y -direction is measured with respect to the horizontal axis through point, P,
The distance between the reference axis and the point P is known.
Equations describing the translational motion:
In the x -direction,
The acceleration for the translational motion is
From the free body diagram, an expression is found for
Differentiate equation
Substitute equation
Substitute the given values, m = 20kg and L = 1.4m in equation
In the y -direction,
The acceleration for the translational motion is
From the free body diagram, an expression is found for
From point, P, there is no vertical displacement.
Differentiate equation
Using the sum of moments find an expression for the reaction force, R.
Mass inertia about the center, G, in the y -direction found using the mass inertia of a hollow cylinder in the y -direction.
Substitute this in the sum of moments equation to find an expression for R.
Substitute equations
Substitute the given values, m = 20kg and L = 1.4m in equation
Conclusion:
Equations of motion of the rod in terms of
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Chapter 3 Solutions
System Dynamics
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