Unit5 - Subjective Questions

Dynamics is the branch of engineering mechanics that deals with the study of bodies in motion. It is further divided into two main branches:

• Kinematics: It is the study of the geometry of motion without considering the forces that cause the motion. It relates the displacement, velocity, acceleration, and time of a moving body.
• Kinetics: It is the study of the relationship between the forces acting on a body, the mass of the body, and the resulting motion. It predicts the motion caused by given forces or determines the forces required to produce a specific motion.

Newton's Laws of Motion are the fundamental principles of dynamics:

• First Law (Law of Inertia): A body continues in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by an external unbalanced force.
• Second Law (Law of Momentum): The rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction in which the force acts. Mathematically, it is expressed as , where is mass and is acceleration.
• Third Law (Action and Reaction): To every action, there is always an equal and opposite reaction. This means forces always occur in pairs.

Rectilinear Motion is the motion of a particle along a straight line.

Derivations for Uniform Acceleration ():
Let initial velocity , final velocity , time , and displacement .

1. Velocity-Time Relation: By definition, acceleration is the rate of change of velocity: . Rearranging gives:
2. Displacement-Time Relation: Average velocity is . Displacement is average velocity multiplied by time: . Substituting gives:
3. Velocity-Displacement Relation: From , we get . Substituting this into gives: . Rearranging yields:

• Translation: A rigid body is in translation if any straight line drawn on the body remains parallel to its original position throughout the motion. All particles of the body have the same displacement, velocity, and acceleration at any given instant. It can be rectilinear (straight path) or curvilinear (curved path).
• Rotation: A rigid body is in rotation if all its particles move in circular paths about a single fixed axis. The particles have different linear velocities and accelerations depending on their distance from the axis of rotation, but the entire body shares the same angular velocity and angular acceleration.

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 as well as an angular displacement simultaneously.

Characteristics:

• The motion can be analyzed by breaking it down into a translation of a reference point (usually the center of mass) and a rotation about that reference point.

Examples:

1. A wheel rolling on a flat surface without slipping. The center of the wheel translates linearly while the rest of the wheel rotates about the center.
2. The motion of a connecting rod in a reciprocating steam or internal combustion engine.

Plane Curvilinear Motion occurs when a particle moves along a curved path that lies entirely within a single plane.

Description of Position:
To describe the motion, a fixed coordinate system is required. The position of a particle at any instant can be defined using:

1. Rectangular Coordinates (): Position vector is .
2. Normal and Tangential Coordinates (): Defined relative to the path itself, where the tangential axis is tangent to the curve in the direction of motion, and the normal axis is perpendicular to the tangent, pointing towards the center of curvature.
3. Polar Coordinates (): Defined by the radial distance from a fixed origin and an angular position .

In plane curvilinear motion, it is often convenient to resolve acceleration into components parallel and perpendicular to the path:

• Tangential Acceleration (): This component acts along the tangent to the path. It represents the rate of change of the magnitude of the velocity (speed). It is given by:
• Normal Acceleration (): This component acts perpendicular to the tangent, directed towards the center of curvature of the path. It represents the rate of change of the direction of the velocity. It is given by: , where is the speed and is the radius of curvature.
• Total Acceleration (): The vector sum of both components, with magnitude .

Projectile: A particle projected into the air at an angle to the horizontal, moving under the sole influence of gravity (ignoring air resistance), is called a projectile.

Derivation of Maximum Height ():
Let a particle be projected with initial velocity at an angle to the horizontal.

• The vertical component of initial velocity is .
• The acceleration in the vertical direction is (upward is positive).
• At the maximum height, the vertical component of final velocity is zero ().

Using the kinematic equation:
Substituting the values:

D'Alembert's Principle states that a system of rigid bodies is in dynamic equilibrium if the sum of the external forces and the reversed effective forces (inertia forces) acting on the system is zero.

Mathematically, it is a restatement of Newton's Second Law. From , we can write . The term () is known as the inertia force.

Significance:
By introducing the fictitious inertia force, a dynamic problem (involving acceleration) is reduced to a static problem (equilibrium). This allows engineers to use the familiar equations of statics ( , , ) to solve complex dynamics problems.

These are the fundamental kinematic quantities for rotational motion:

• Angular Displacement (): The angle (usually in radians) through which a body or a point has rotated about a specific axis.
• Angular Velocity (): The rate of change of angular displacement with respect to time. It is given by . Its SI unit is radians per second (rad/s).
• Angular Acceleration (): The rate of change of angular velocity with respect to time. It is given by . Its SI unit is radians per second squared (rad/s).

Consider a particle moving in a circular path of radius . Let the particle move from position A to position B in a small time interval , covering a linear distance (arc length) and an angular displacement .

From geometry, the arc length is related to the angle by:
Dividing both sides by the time interval :
Taking the limit as :
(Linear Velocity)
(Angular Velocity)

Therefore, substituting the limits yields the relationship:

Distance vs. Displacement:

• Distance is a scalar quantity representing the total length of the actual path covered by a moving particle, regardless of direction.
• Displacement is a vector quantity representing the shortest straight-line distance between the initial and final positions of the particle, with a specific direction.

Speed vs. Velocity:

• Speed is a scalar quantity representing the rate at which distance is covered (Speed = Distance / Time).
• Velocity is a vector quantity representing the rate of change of displacement (Velocity = Displacement / Time). It specifies both the magnitude (speed) and the direction of motion.

A velocity-time () graph plots velocity on the y-axis and time on the x-axis.

• Determining Displacement: The displacement of a particle over a given time interval is equal to the area under the velocity-time curve for that interval. Mathematically, . If the curve goes below the time axis, that area represents negative displacement.
• Determining Acceleration: The acceleration at any given instant is equal to the slope of the tangent to the velocity-time curve at that instant. Mathematically, . For uniform acceleration, the slope is constant, and the graph is a straight line.

• Uniform Acceleration: A body has uniform acceleration if its velocity changes by equal amounts in equal intervals of time, no matter how small the time intervals are. The acceleration magnitude and direction remain constant.
  • Example: A freely falling body under the influence of gravity ().
• Non-Uniform (Variable) Acceleration: A body has non-uniform acceleration if its velocity changes by unequal amounts in equal intervals of time, or if the direction of acceleration changes.
  • Example: A car moving through traffic, speeding up and slowing down randomly, or a car moving at a constant speed along a curved path (direction of velocity changes, hence acceleration exists and changes direction).

Let a particle be projected from the origin ($0,0$) with initial velocity at an angle to the horizontal.

Horizontal motion:
Velocity remains constant ().
Horizontal displacement at time : (Equation 1)

Vertical motion:
Initial vertical velocity (), acceleration ().
Vertical displacement at time : (Equation 2)

Substitute Equation 1 into Equation 2:

Since , , and are constant, this equation is of the form , which is the equation of a parabola. Thus, the trajectory of a projectile is a parabola.

Horizontal Range () is the horizontal distance covered by the projectile during its time of flight ().
Time of flight is derived from setting vertical displacement :

The horizontal velocity remains constant at .
Horizontal Range = Horizontal Velocity Time of Flight


Using the trigonometric identity :

Condition for Maximum Range:
The range is maximum when .

The maximum range is .

Relative motion analyzes the motion of a particle relative to a moving frame of reference rather than a fixed (absolute) frame.

Consider two particles, A and B, moving in space. Let their position vectors with respect to a fixed origin be and .
If an observer is stationed on particle A, the position of B as seen from A is the relative position vector .

Vector addition gives:
Differentiating with respect to time yields the relative velocity equation:

Differentiating again yields the relative acceleration equation:

This concept is crucial for solving problems where objects move simultaneously, such as aircraft in flight or moving vehicles on a road.

For constant angular acceleration ():
By definition, angular acceleration is:
We can apply the chain rule to eliminate time ():

Since angular velocity , we get:

Separating variables:

Integrating both sides from initial state () to final state ():



Rearranging gives the final equation:

For a particle moving in a circular path of constant radius , the acceleration has two components:

1. Tangential Acceleration (): It is directed along the tangent to the circular path. It is responsible for the change in the magnitude of linear velocity (speed).
   
   where is the angular acceleration.

2. Normal or Centripetal Acceleration (): It is directed along the radius towards the center of the circle. It is responsible for changing the direction of the velocity vector.
   

The total acceleration is the vector sum of these orthogonal components: .

Given the position function:

1. Velocity (): Velocity is the first derivative of displacement with respect to time ().
   
   

2. Acceleration (): Acceleration is the first derivative of velocity with respect to time ().
   
   

These expressions allow the calculation of velocity and acceleration at any given instant of time .