Dynamics

Scalar & vector quatities

• Scalar - A physical quantity which has magnitude ONLY.
• Vector - A physical quantity which has both magnitude and direction

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Addition & Subtraction of Scalar Quantities

Two or more scalar quantities can be added if:

They are the same physical quantities

The sum (or difference) of the scalar quantities is the algebraic sum (or difference) of the magnitude of the scalar quantities.

Distance, x

• Is a Scalar Quantity, defined as the length between 2 points,
• The units of distance is meters (m)
• Distance = speed x time
• Area of Velocity Time Graph is Distance

Displacement, x

Is a vector quantity. Defined as the distance travelled in a specific direction. The unit of displacement is the meter.

Guidelines for determining the signs of Displacement

• Rightward displacement is considered positive.
• Leftward displacement is considered negative.
• Upward displacement is considered positive.
• Downward displacement is considered negative.

Speed (u, v)

Speed is a vector quantity that represents the rate of change of position in relation to time. It can be computed by dividing the difference in position by the corresponding difference in time.

Speed (u, v) = Position difference (Δx) / Time difference (Δt)

Speed (u, v) = Δx / Δt

The measurement unit for speed is meters per second (m/s), encompassing both magnitude and direction.

Average speed

Average speed is a vector quantity obtained by dividing the total position change by the total time elapsed.

Average speed = Total position change (Δx) / Total time elapsed (Δt)

Average speed = Δx / Δt

The unit of average speed is also meters per second (m/s), considering both magnitude and direction.

Velocity

Velocity is a vector quantity and is defined as the ratio of change of displacement (speed & direction) with respect to time.

Velocity (u, v)= change in displacement(Δx)/time taken (t)

Velocity (u, v) = Δx/t

Unit of Velocity is ms^-1 or meters/seconds

Guidelines for determining the signs of Velocity

1. Velocity to RIGHT is POSITIVE
2. Velocity to LEFT is NEGATIVE
3. Velocity UPWARD is POSITIVE
4. Velocity DOWNWARD is NEGATIVE

Acceleration

Acceleration is a vector quantity and is defined as the rate of change of velocity with respect to time.

Acceleration (a) = change in velocity (Δv)/time taken(t)

Acceleration (a) = Δt/t

Unit is ms^-2 or meters/seconds^2

If a particle undergoes acceleration...

1. Direction is constant, but speed is changing with respect to time.
2. Speed is constant, but direction is changing with respect to time.
3. Both speed & direction are changing with respect to time.

• A POSITIVE acceleration implies and INCEASE in speed
• A NEGATIVE acceleration implies and DECEASE in speed

A NEGATIVE acceleration is called a deceleration

Graphs

Displacement Graph

Velocity-time Graph

Linear Momentum

Linear momentum is a vector quantity that describes the motion of an object and is calculated by multiplying its mass (m) by its velocity (v).

Linear Momentum (p) = mass (m) × velocity (v)

p = m × v

The unit of linear momentum is kilogram-meter per second (kg·m/s).

Principle of Linear Momentum Conservation

In interactions between objects, such as collisions or explosions, linear momentum is conserved when no external forces are applied to the system. This principle states that the total linear momentum before the interaction is equal to the total linear momentum after the interaction.

Inertia

The concept of inertia is associated with the mass of an object. Inertia refers to the property of an object to resist a change in its state of motion, whether it is at rest or already in motion.

Forces

A force can be described as a push or a pull, or any action that has the potential to modify the motion of an object or alter its shape.

Formulas

Based on Newton's Second Law:

Force = Rate of change of momentum

Force = (mv - mu) / t

Force F = m(v - u) / t

Here, (v - u) represents the change in velocity

(v - u) / t represents the change in velocity over time, which is equivalent to acceleration

Force = mass × acceleration

F = m × a

Change in Momentum

The change in momentum can be found by subtracting the initial momentum (p₁) from the final momentum (p₂).

Change in Momentum (Δp) = p₂ - p₁

Newton's Second Law

According to Newton's Second Law, force (F) is equal to the rate of change of momentum.

Force = Rate of change of momentum

Force = Δp / t

Here, Δp represents the change in momentum, and t represents the time interval.

Impulse

Impulse is synonymous with the change in momentum and can be calculated by multiplying the force (F) by the time interval (t).

Impulse (I) = F * t

Equations

1. Impulse (I) = Change in Momentum
2. Impulse = Force * time

Newton's Laws of Motion

The principles of Newton's Laws of Motion dictate the behavior of objects in motion. As per the first law, objects at rest remain stationary, while objects in motion persist in moving along a straight path unless acted upon by an external force. The second law establishes a proportional relationship between force and the rate of momentum change in the absence of external forces. Lastly, the third law asserts that every action corresponds to an equal and opposite reaction, whereby the force exerted by one body on another is reciprocated with an equal force in the opposite direction.

Newton's 2nd Law

Rate of change of momentum = Change in momentum/time taken

Initial momentum = Mu

Final Momentum = Mv

Change in momentum = final momentum - initial momentum

= Mv - Mu

Rate of change of momentum = Mv - Mu/t

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