FSTM 3094 Principles of Mechanical Science Notes

FSTM 3094

PRINCIPLES OF MECHANICAL SCIENCE

Chapter 3: MOTION (ONE AND TWO DIMENSIONS) AND ROTATIONAL MOTION

1. GRAVITY AND FREE-FALLING BODIES

• Definition: Gravity is defined as the constant rate at which an object accelerates while falling downwards. This applies to objects being tossed either upwards or downwards, and assumes the elimination of air resistance on flight.

• Magnitude: The gravitational acceleration is represented by the symbol g.

  • $g = 9.8 \, m/s^2$

  • $g = 32 \, ft/s^2$

• Independence of Mass: The acceleration due to gravity is the same for all objects regardless of their mass, density, or shape.

• Direction of Motion:

  • Motion is described along a vertical (y) axis.

  • The positive direction of y is upward.

  • The free-fall acceleration is considered negative, as it acts downwards toward the Earth’s center, thus represented as –g in equations.

• Equations of Motion for Free-Falling Bodies:

  1. $v = u - gt$

  2. $v^2 = u^2 - 2gs$

  3. $s = ut - rac{1}{2}gt^2$

  4. $s = vt + rac{1}{2}at^2$

  5. $s = rac{(u + v)}{2}t$

2. PROJECTILE MOTION

Learning Outcome

• Be able to explain the magnitudes and directions of components involved in projectile motion.

Definition of Projectile Motion

• Defined as motion of a particle in a vertical plane with some initial velocity, u, where the only acceleration acting upon it is the free-fall acceleration, which acts downward.

• Projection Assumption: It is assumed that air resistance has a negligible effect on the projectile.

• Horizontal Motion: There is no acceleration in the horizontal direction.

• Vertical Motion: Experiences constant acceleration due to gravity.

Components of Projectile Motion

• Horizontal Component:

  • Acceleration, $a = 0 \, m/s^2$

  • Initial Velocity: $u_x = u \, ext{cos}(θ)$

  • Velocity Function: $V_x = u_x + a_xt$

  • Displacement Function: $S_x = u_xt$

• Vertical Component:

  • Acceleration: $a_y = -g = -9.81 \, m/s^2$

  • Initial Velocity: $u_y = u \, ext{sin}(θ)$

  • Velocity Function: $V_y = u_y - gt$

  • Displacement Function: $S_y = u_yt - rac{1}{2}gt^2$

Real-Life Applications of Projectile Motion

• Practical implications of understanding projectile motion include applications in sports, engineering, and various physics simulations.

3. ROTATIONAL KINEMATICS

• Definition of Rigid Body: A rigid body is defined as one that can rotate such that all parts of the body remain fixed together without changing its shape.

• Axis of Rotation: Rotation occurs around a fixed axis.

• Examples of Rigid Bodies: Include wheels, shafts, pulleys, and gyroscopes.

4. ANGULAR DISPLACEMENT

• Definition: Angular displacement describes the amount of rotation about an axis.

• Unit of Measurement: The radian (rad).

• Radian Definition:

  • $θ = rac{s}{R}$

  • where s is the arc length and R is the radius of the circle.

• Conversions:

  • 1 complete revolution = 360° = $2 ext{π}$ radians.

• Angular Displacement Formula:

  • If rotating about an axis, changing the angular position from $θ_1$ to $θ_2$ results in an angular displacement of: $ext{Δ}θ = θ_2 - θ_1$

• Direction of Angular Displacement: Positive for counterclockwise and negative for clockwise movements.

5. ANGULAR VELOCITY

• Definition: The rate of change of angular displacement.

• Formula:

  • $ω = rac{θ_2 - θ_1}{t_2 - t_1} = rac{ ext{Δ}θ}{ ext{Δ}t}$

• Unit of Angular Velocity: Standard units include radians per second (rad/s) and revolutions per second (rev/s).

• Sign of Angular Velocity:

  • Angular velocity, ω, is positive for counterclockwise rotation and negative for clockwise rotation.

6. ANGULAR ACCELERATION

• Definition: Occurs when the angular velocity of a body is not constant.

• Formula:

  • $α = rac{ω_2 - ω_1}{t_2 - t_1} = rac{Δω}{Δt}$

• Units of Angular Acceleration: Commonly expressed in radians per second squared (rad/s²) or revolutions per second squared (rev/s²).

7. RELATIONSHIP BETWEEN ROTATIONAL AND LINEAR MOTION

• Definition of Axis of Rotation: The line of particles that remain stationary during rotation.

• Linear Velocity: The greater the distance a particle is from the axis of rotation, the greater the linear velocity.

• Linear Velocity Formula:

  • $v = ωR$

• Tangential Acceleration:

  • Given by $a_T = αR$

• Centripetal Acceleration: Defined as:

  • $a_c = rac{v^2}{R}$ or $a_c = rω^2$

• Difference Between Accelerations:

  • Tangential acceleration represents a change in linear velocity; centripetal acceleration represents a change in direction of motion.

• Resultant Acceleration: Computed from the vector sum of both tangential and centripetal acceleration.

8. SYMBOLS IN LINEAR AND ROTATIONAL MOTION

9. COMPARISONS BETWEEN LINEAR AND ROTATIONAL MOTION EQUATIONS

• Equations of Linear Motion with Uniform Acceleration:

  1. $v = u + at$

  2. $v^2 = u^2 + 2as$

  3. $s = ut + rac{1}{2}at^2$

• Equations of Rotational Motion with Uniform Angular Acceleration:

  1. $ω = ω_1 + αt$

  2. $ω^2 = ω_1^2 + 2αΔθ$

  3. $Δθ = ω_ot + rac{1}{2}αt^2$

10. MOMENT OF INERTIA

• Definition: Moment of inertia (I) is also known as rotational inertia. It is a measure of an object's resistance to changes in its angular motion.

• Formula:

  • $I = mr^2 + mr^2 + m_2r^2 + … + m_nr^2 = Σmr^2$

• Units: Measured in kilogram meter squared (kg m²).

11. KINETIC AND KINETIC ENERGY

• Kinetic Energy (KE): The energy an object possesses due to its motion, which increases with the square of velocity.

• Relation to rotational motion can be derived analogously to linear motion equations, considering the radius from the axis of rotation, etc. -