Unit6 - Subjective Questions

D'Alembert's Principle:

D'Alembert's principle states that the system of external forces acting on a body in motion is in dynamic equilibrium with the inertia forces of the body.

• Mathematical Representation: The equation of motion, , can be rewritten as .
• Inertia Force: The term is called the inertia force. It acts through the center of mass and is directed opposite to the acceleration.
• Plane Motion Application: For a rigid body in plane motion, the principle is extended to include rotational motion. The external forces and moments are in equilibrium with the inertia force and the inertia couple , where is the acceleration of the mass center, is the mass moment of inertia about the mass center, and is the angular acceleration.
• Equations:

Kinetics of Pure Rotation using D'Alembert's Principle:

Consider a rigid body rotating about a fixed axis passing through point O with an angular acceleration .

• Kinematics: The acceleration of the center of mass G (at distance from O) has two components: tangential and normal .
• Inertia Forces and Couples:
  1. Tangential inertia force:
  2. Normal inertia force:
  3. Inertia couple:
• Dynamic Equilibrium: By D'Alembert's principle, the external forces and the inertia forces/couples form a system in equilibrium.
• Taking moments about the axis of rotation O:
  
  
  
• By parallel axis theorem, . Therefore:
  
  This is the fundamental equation of motion for a rigid body in pure rotation.

Work:
Work is done by a force when its point of application moves through a displacement. For a rigid body subject to a system of forces and moments, the total work done from position 1 to position 2 is the sum of the work done by the forces and the couples.

Kinetic Energy (KE):
The kinetic energy of a rigid body in general plane motion is the sum of its translational kinetic energy (due to the velocity of its mass center) and its rotational kinetic energy (due to its angular velocity about the mass center).

• 
  Where:
• = mass of the body
• = velocity of the center of gravity
• = mass moment of inertia about the center of gravity
• = angular velocity of the body

Work-Energy Principle:
The principle states that the total work done by all external forces and couples acting on a rigid body is equal to the change in its kinetic energy.

Proof:
Consider the equations of motion for a rigid body:

Integrating the force equation with respect to displacement :

Integrating the moment equation with respect to angular displacement :

Adding both gives the total work done:


Hence proved.

Applying D'Alembert's Principle to Connected Bodies:
When two or more bodies are connected (e.g., by strings over pulleys), their motions are kinematically related.

Steps:

1. Kinematic Relationship: Establish the relationship between the accelerations of the connected bodies. For example, if two blocks are connected by an inextensible string, they share the same magnitude of acceleration, . If a pulley is involved, relate its angular acceleration to the linear acceleration ().
2. Free Body Diagrams (FBD): Draw a separate FBD for each body, showing all external forces (gravity, normal reaction, tension, friction).
3. Inertia Forces: Apply D'Alembert's principle by introducing pseudo-forces (inertia forces). For a translating body, apply an inertia force opposite to its acceleration. For a rotating body, apply an inertia couple opposite to its angular acceleration.
4. Dynamic Equilibrium: Write the equilibrium equations (, , ) for each body including the inertia terms.
5. Solve: Solve the simultaneous equations to find the unknown accelerations and internal forces (like tension in strings).

Inertia Couple:

• Definition: In D'Alembert's principle, just as a translating body resists linear acceleration with an inertia force (), a rotating rigid body resists angular acceleration with an inertia couple (or inertia moment).
• Magnitude and Direction: The magnitude of the inertia couple is , where is the mass moment of inertia about the center of mass, and is the angular acceleration. Its direction is strictly opposite to the direction of the angular acceleration .
• Role in Plane Motion: In general plane motion, the body undergoes both translation and rotation. To reduce this dynamic state to a static equilibrium state (D'Alembert's approach), one must apply both an inertia force vector ( through the center of mass) and an inertia couple ().
• Equation: , showing that the sum of external moments about the mass center plus the inertia couple equals zero.

Comparison:

1. Fundamental Basis:
   • D'Alembert's Principle: Based on Newton's Second Law. Converts a dynamic problem into an equivalent static equilibrium problem by adding inertia forces.
   • Work-Energy Principle: Based on the integration of Newton's Second Law over distance. Relates the work done by forces to the change in kinetic energy.
2. Variables Involved:
   • D'Alembert's: Deals with forces, masses, and accelerations at a specific instant.
   • Work-Energy: Deals with forces, masses, velocities, and displacements over an interval.
3. Use Cases:
   • D'Alembert's: Best suited for finding instantaneous accelerations, reaction forces, and tensions at a specific moment.
   • Work-Energy: Best suited for problems involving changes in velocity over a certain distance, especially when time and acceleration are not explicitly required.
4. Vector vs. Scalar:
   • D'Alembert's: Involves vector equations (forces and moments have directions).
   • Work-Energy: Involves scalar equations (energy and work are scalar quantities), which often simplifies the math.

Kinetics of Translation:
Translation is a motion where every line segment in the body remains parallel to its original direction during the motion. There is no rotation (). Translation can be rectilinear (straight line) or curvilinear (curved path).

Equations of Motion:
Since there is no angular acceleration (), the inertia couple is zero. The body can be treated as a particle located at its center of mass .

Key Concept: Even though the body is not rotating, the external forces must be arranged such that their resultant moment about the center of mass is zero, otherwise, the body would rotate. Using D'Alembert's principle, we apply the inertia forces and through the center of gravity to establish dynamic equilibrium.

Formulation:

1. Forces acting on the cylinder:
   • Weight acting downwards from the center of mass (G).
   • Normal reaction perpendicular to the incline.
   • Frictional force acting up the incline (causes rolling).
2. Kinematics (Rolling without slipping):
   • Linear acceleration , where is the angular acceleration.
3. Applying D'Alembert's Principle:
   • Assume a coordinate system with x-axis down the incline and y-axis perpendicular to it.
   • Apply inertia force up the incline through G.
   • Apply inertia couple counter-clockwise.
   • for a solid cylinder.
4. Equilibrium Equations:
   • (Eq. 1)
   • (Eq. 2)
5. Solving:
   Substitute into Eq. 1:
   
   
   

Application of Work-Energy Principle to Connected Bodies:

When a system consists of multiple interconnected rigid bodies (e.g., blocks connected by cables, gear trains), the work-energy principle can be applied to the entire system as a single entity.

• System Approach: The principle states:
• Total Work Done (): This includes the work done by all external forces (gravity, applied forces) acting on all bodies in the system.
  • Crucial Point: The work done by internal forces (like tension in an inextensible connecting string) cancels out. The work done by tension on one body is positive, and on the other is negative, summing to zero.
• Total Kinetic Energy (): The kinetic energy of the system is the sum of the kinetic energies of the individual bodies.
• Kinematic Constraints: The velocities () and angular velocities () of the individual bodies are related through the system's kinematic constraints (e.g., for a rope unwinding from a pulley). These relations must be used to express all kinetic energies in terms of a single variable to solve the equation.

General Plane Motion:
General plane motion is a complex motion of a rigid body that is a combination of both translation and rotation. In this motion, the body undergoes a linear displacement while simultaneously changing its angular orientation in the same plane.

Characteristics:

• The displacement of the body can be separated into a translation of a specific point (usually the center of mass) and a rotation of the body about that point.
• Velocity:
• Acceleration: (where relative acceleration includes tangential and normal components).

Examples:

1. A rolling wheel: A wheel rolling on a road translates forward while rotating about its axle.
2. Connecting rod in an engine: The connecting rod translates up and down with the piston while simultaneously rotating to accommodate the crank's circular motion.
3. A sliding ladder: A ladder sliding down a wall undergoes translation (center of mass moves down and out) and rotation (the angle changes).

Derivation of Kinetic Energy for Plane Motion:

Consider a rigid body in plane motion. Let be the center of mass, moving with velocity , and let the body rotate with angular velocity .

1. Consider a small mass element at a position vector relative to the center of mass .
2. The absolute velocity of the mass element is , where .
3. The kinetic energy of this small mass element is .
   
   
4. Integrate over the entire mass of the body:
   
5. Simplifying the terms:
   • (Total mass)
   • (Mass moment of inertia about )
   • The term because is measured from the center of mass .
6. Substituting these into the equation:
   
   This shows the total kinetic energy is the sum of translational KE of the mass center and rotational KE about the mass center.

Work-Energy Method with Springs:

When a linear spring connects to a rigid body in a system, the spring exerts a variable force , where is the spring stiffness and is the deformation (extension or compression) from its unstretched length.

• Work Done by a Spring: The work done by a spring on a system as it moves from deformation to is equal to the negative change in its elastic potential energy.
  
• Application in the Equation: In the work-energy equation , the work done by the spring must be included in .
• Alternative Method (Conservation of Energy): If only conservative forces (like gravity and springs) do work, it's easier to use the conservation of energy principle:
  
  Where is the elastic potential energy. The spring stores energy as it deforms and releases it as it returns to its unstretched state.

Rotation about an Eccentric Axis:

When a rigid body rotates about a fixed axis that does not pass through its center of mass (distance ), both translation of the center of mass and rotation of the body occur.

1. Kinematics of Center of Mass: The center of mass travels in a circular path of radius . Its accelerations are:
   • Tangential:
   • Normal:
2. Equations of Motion (Newton-Euler):
3. Alternative Moment Equation (using D'Alembert or parallel axis):
   Taking moments about the fixed pivot point :
   
   Where is the mass moment of inertia about axis O. By the parallel-axis theorem, .
4. Reaction Forces: The equations and are primarily used to determine the dynamic reaction forces at the pin/hinge support .

Calculating Support Reactions using D'Alembert's Principle:

1. Identify Kinematics: Determine the linear acceleration of the center of mass and the angular acceleration of the rigid body from given kinematic conditions or constraints.
2. Free Body Diagram (FBD): Draw the FBD of the rigid body, replacing supports with their respective unknown reaction forces (e.g., a pin support has and , a roller has a normal force ). Show all applied active forces (gravity, external loads).
3. Apply Inertia Vectors: Add the inertia forces ( and ) acting through the center of mass, and the inertia couple (). Ensure these are directed opposite to the respective assumed positive accelerations.
4. Static Equilibrium: With the inertia terms included, the body is treated as being in static equilibrium. Apply the three planar equilibrium equations:
   • (Sum of external forces in x + inertia force in x = 0)
   • (Sum of external forces in y + inertia force in y = 0)
   • (Sum of moments about any convenient point A + moment of inertia forces about A + inertia couple = 0).
   • Tip: Choosing point A as a support location often eliminates unknown reaction forces from the moment equation, making it easier to solve.

Derivation:
Let . moves down with acceleration , moves up with . Pulley rotates with angular acceleration .
Tensions on the two sides of the pulley are different, say (side ) and (side ).

1. Using D'Alembert's Principle / Newton's 2nd Law:

• For block 1 (): Moving down.
  Equation:
• For block 2 (): Moving up.
  Equation:
• For the pulley (): Rotating. (assuming uniform disk).
  Equation:
  

2. Solving for acceleration :
Substitute and into the pulley equation:




This provides the linear acceleration of the system considering the inertia of the rotating pulley.

Mass Moment of Inertia ():

• Definition: Mass moment of inertia is a measure of an object's resistance to changes in its rotation rate. It is the rotational analog of mass for linear motion. Mathematically, it is the integral of the second moment of mass with respect to an axis: , where is the perpendicular distance from the axis of rotation to the mass element .
• Significance:
  1. Governs Rotational Acceleration: In the equation , for a given applied torque (), a larger mass moment of inertia results in a smaller angular acceleration ().
  2. Depends on Mass Distribution: Unlike linear mass, depends not just on the total mass, but how that mass is distributed relative to the axis of rotation. Mass located further from the axis contributes significantly more to (due to the term).
  3. Kinetic Energy: It determines the rotational kinetic energy of a body (). Flywheels are designed with a large mass moment of inertia to store significant rotational energy.

Simplification via Work-Energy Principle:

1. Handling of Friction in Pure Rolling: In pure rolling (rolling without slipping), the instantaneous point of contact between the rigid body and the surface has zero velocity. Therefore, the static friction force responsible for rolling does no work.
2. Work-Energy Advantage: Since the frictional force does zero work, it does not appear in the Work-Energy equation (). We only need to account for conservative forces (like gravity) doing work. This directly links the change in height (potential energy/work done by gravity) to the change in velocity (kinetic energy) without needing to solve for the frictional force.
3. D'Alembert's Approach Requirement: In D'Alembert's principle (or Newton-Euler equations), the frictional force must be explicitly included in the force equilibrium equations (). You must set up simultaneous equations combining linear equilibrium, moment equilibrium (), and the kinematic constraint () to solve for the acceleration, often calculating the frictional force as an intermediate step.
4. Conclusion: The work-energy principle often yields velocities directly and more rapidly by bypassing the calculation of internal forces and static friction in non-slipping constraints.

Work Done by a Couple:

• When a rigid body is subjected to a couple with moment , and it undergoes an angular displacement in the plane of the couple, the infinitesimal work done is .
• The total work done during a finite rotation from an initial angle to a final angle is given by the integral:
  
• Constant Moment: If the couple moment is constant throughout the rotation, the integration simplifies to:
  
• Sign Convention: The work done is positive if the couple moment and the angular displacement are in the same direction (e.g., both clockwise or both counter-clockwise). It is negative if they are in opposite directions.

Parallel Axis Theorem:
The parallel axis theorem states that the mass moment of inertia of a body about any axis () is equal to its mass moment of inertia about a parallel axis passing through its center of mass () plus the product of the mass of the body () and the square of the perpendicular distance () between the two axes.

• Formula:

Essential Role in Plane Motion:

1. Rotation about Eccentric Axes: In kinematics of rigid body rotation, bodies often rotate about hinges or pivots that are not at their center of mass (e.g., a pendulum). The equation of motion is . The theorem allows for easy calculation of from the standard tabulated values of .
2. General Plane Motion Analysis: In methods analyzing instantaneous centers of zero velocity (IC), the body is treated as if it is in pure rotation about the IC at that specific instant. To use the rotational kinetic energy formula or the dynamic equation , the parallel axis theorem must be used to find the moment of inertia about the IC ().