Mechanics - CSEC Physics

Introduction to Mechanics

Mechanics is the branch of physics that deals with the motion of objects and the forces that affect that motion.

Scalars and Vectors

Scalar Quantities

Have magnitude only (no direction):

Vector Quantities

Have both magnitude and direction:

Scalar: 50 km/h 50 Vector: 50 km/h North N 50 km/h

Motion

Displacement vs Distance

Speed vs Velocity

Average speed = Total distance / Total time

Average velocity = Displacement / Time

Example: Speed and Velocity

A car travels 60 km north in 1 hour, then 40 km south in 0.5 hour. Calculate:

a) Total distance traveled

b) Average speed

c) Displacement

d) Average velocity

Solution:

a) Distance = 60 + 40 = 100 km

b) Average speed = 100 km / 1.5 h ≈ 66.67 km/h

c) Displacement = 60 km N - 40 km S = 20 km N

d) Average velocity = 20 km N / 1.5 h ≈ 13.33 km/h N

Acceleration

Rate of change of velocity with time:

a = (v - u) / t

Where:

Velocity (m/s) Time (s) Zero acceleration Positive acceleration Negative acceleration

Equations of Motion

1. v = u + at

2. s = ut + ½at²

3. v² = u² + 2as

Where:

Example: Equations of Motion

A car accelerates from rest at 3 m/s² for 8 seconds. Calculate:

a) Final velocity

b) Distance traveled

Solution:

Given: u = 0 m/s, a = 3 m/s², t = 8 s

a) v = u + at = 0 + (3 × 8) = 24 m/s

b) s = ut + ½at² = 0 + ½ × 3 × 8² = 96 m

Forces and Newton's Laws

Newton's First Law (Law of Inertia)

An object remains at rest or in uniform motion unless acted upon by an external force.

Newton's Second Law

F = ma

Where:

Newton's Third Law

For every action, there is an equal and opposite reaction.

Fapplied Ffriction Fgravity Fnormal

Example: Newton's Second Law

What force is needed to accelerate a 5 kg object at 4 m/s²?

Solution:

F = ma = 5 × 4 = 20 N

Weight and Mass

Weight = mass × acceleration due to gravity

W = mg

Where:

Note: Mass is constant (amount of matter), while weight depends on gravity (a force).

Friction

The force that opposes relative motion between two surfaces in contact.

Types of Friction

Frictional force (Ffriction) = μ × Normal force (Fnormal)

Where μ (mu) is the coefficient of friction (depends on surfaces)

Work, Energy, and Power

Work

Work = Force × Displacement × cosθ

W = Fd cosθ

Where θ is the angle between force and displacement vectors

Energy

Capacity to do work. Types include:

KE = ½mv²

PE = mgh

Power

Rate of doing work:

Power = Work / Time

P = W/t

Example: Energy Calculations

A 2 kg object is dropped from 10 m height. Calculate:

a) Potential energy at top

b) Velocity just before impact

Solution:

a) PE = mgh = 2 × 9.8 × 10 = 196 J

b) At impact, all PE → KE:

½mv² = 196 → v² = (2 × 196)/2 = 196 → v = √196 = 14 m/s

Momentum and Impulse

Momentum

Momentum (p) = mass × velocity

p = mv

Impulse

Change in momentum:

Impulse = Force × Time = Change in momentum

Ft = Δp = mΔv

Conservation of Momentum

In a closed system with no external forces, total momentum before collision equals total momentum after collision.

Example: Momentum Conservation

A 5 kg object moving at 6 m/s collides with a stationary 3 kg object. If they stick together, find their final velocity.

Solution:

Initial momentum = 5 × 6 + 3 × 0 = 30 kg·m/s

Final momentum = (5 + 3) × v = 8v

Conservation: 8v = 30 → v = 30/8 = 3.75 m/s

Projectile Motion

Motion of objects projected into the air, influenced by gravity.

Key Characteristics

vx vy v Projectile Path

Circular Motion

Centripetal Force

F = mv²/r

Where:

Centripetal Acceleration

a = v²/r

r v F

Glossary of Terms

Acceleration
Rate of change of velocity with time.
Force
A push or pull that can cause an object to accelerate.
Friction
Force that opposes relative motion between surfaces.
Inertia
Tendency of an object to resist changes to its motion.
Kinetic Energy
Energy possessed by a moving object.
Momentum
Product of an object's mass and velocity.
Potential Energy
Stored energy due to an object's position.
Scalar
Quantity with magnitude only.
Vector
Quantity with both magnitude and direction.
Velocity
Speed in a given direction.

Self-Assessment Questions

  1. A car accelerates from 20 m/s to 30 m/s in 5 seconds. Calculate its acceleration.

    2. a = (v - u)/t = (30 - 20)/5 = 2 m/s²

  2. Calculate the weight of a 10 kg object on Earth (g = 9.8 m/s²).

    3. W = mg = 10 × 9.8 = 98 N

  3. A force of 50 N acts on a 5 kg mass. Calculate the acceleration produced.

    4. F = ma → a = F/m = 50/5 = 10 m/s²

  4. A ball is thrown vertically upward at 20 m/s. Calculate its maximum height (g = 9.8 m/s²).

    5. At max height, v = 0

    v² = u² + 2as → 0 = 20² + 2(-9.8)s → s = 400/19.6 ≈ 20.41 m

  5. Calculate the kinetic energy of a 2 kg object moving at 3 m/s.

    6. KE = ½mv² = ½ × 2 × 3² = 9 J

  6. A 1000 kg car moving at 20 m/s brakes to a stop in 5 seconds. Calculate the braking force.

    7. a = (0 - 20)/5 = -4 m/s²

    F = ma = 1000 × -4 = -4000 N (negative sign indicates opposing force)

  7. A 4 kg object moving east at 6 m/s collides with a 2 kg object moving west at 4 m/s. If they stick together, find their final velocity.

    8. Let east be positive:

    Initial momentum = (4 × 6) + (2 × -4) = 24 - 8 = 16 kg·m/s

    Final mass = 4 + 2 = 6 kg

    Final velocity = 16/6 ≈ 2.67 m/s east

  8. Calculate the work done by a 20 N force pushing an object 5 m in the direction of the force.

    9. W = Fd = 20 × 5 = 100 J

  9. A stone is whirled in a vertical circle at 4 m/s with a radius of 0.5 m. Calculate the centripetal acceleration.

    10. a = v²/r = 4²/0.5 = 16/0.5 = 32 m/s²

  10. A 60 kg person runs up a flight of stairs 5 m high in 10 seconds. Calculate their power output.

    11. Work done = PE gained = mgh = 60 × 9.8 × 5 = 2940 J

    Power = Work/time = 2940/10 = 294 W