
Concept explainers
(a)
The equation of motion of the scale.

Answer to Problem 3.39P
Explanation of Solution
Given:
Friction of the pivot and mass of the scale arm are neglected.
Concept used:
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
Let the angular displacement be
The angular velocity,
Hence, the equation of motion of this object can be rewritten by substituting,
Derivation of Equation of motion:
Free body diagram:
To find the equation of motion, the required unknowns are
The mass moment of Inertia, I about a specified reference axis is given as:
Where r = distance from the reference axis to mass element
Mass moment of Inertia of a rotating pendulum =
Inertia of mass,
Inertia of mass,
Total Inertia,
Sum of moments about the point, A:
Conclusion:
The equation of motion of the scale is
(b)
The equilibrium relation between weight and angle.

Answer to Problem 3.39P
Explanation of Solution
Given:
The system is in equilibrium.
Concept used:
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
Let the angular displacement be
The angular velocity,
Hence, the equation of motion of this object can be rewritten by substituting,
Derivation of mg at equilibrium:
Free body diagram:
To find the equation of motion, the required unknowns are
The mass moment of Inertia, I about a specified reference axis is given as:
Where r = distance from the reference axis to mass element
Mass moment of Inertia of a rotating pendulum =
Inertia of mass,
Inertia of mass,
Total Inertia,
Sum of moments about the point, A:
At equilibrium, the angular acceleration,
Substitute the value of
Conclusion:
When system is at equilibrium, the relation between mg and
(c)
The value of weightof the object.

Answer to Problem 3.39P
22. 713N.
Explanation of Solution
Given:
The mass of counterweight,
Concept used:
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
Let the angular displacement be
The angular velocity,
Hence, the equation of motion of this object can be rewritten by substituting,
Derivation of mg at equilibrium:
Free body diagram:
To find the equation of motion, the required unknowns are
The mass moment of Inertia, I about a specified reference axis is given as:
Where r = distance from the reference axis to mass element
Mass moment of Inertia of a rotating pendulum =
Inertia of mass,
Inertia of mass,
Total Inertia,
Sum of moments about the point, A:
At equilibrium, the angular acceleration,
Substitute the value of
Substitute the given values in equation
Conclusion:
The weight of the object is 22. 713N.
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Chapter 3 Solutions
System Dynamics
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