Chapter 8 - Momentum and Forces

8.1 Momentum and Conservation of Momentum

Momentum

Momentum - relates to its mass and velocity

p = mv

p is momentum (kg m s⁻¹)

m is the mass of the object (kg)

v is the velocity of the object (m s⁻¹)

• Greater the mass/velocity, the larger the momentum

• Vector quantity

Conservation of Momentum

• Momentum is conserved in any interaction or collision between objects

∑p_before = ∑p_after

where ∑p is the sum of the momentum of objects in a system.

Momentum in One dimensional collisions

If two objects are colliding in 1D, this is the equation.

∑mv_before = ∑mv_after

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

m₁ is the mass of object 1 (kg)

u₁ is the initial velocity of object 1 (m s⁻¹)

v₁ is the final velocity of object 1 (m s⁻¹)

m₂ is the mass of object 2 (kg)

u₂ is the initial velocity of object 2 (m s⁻¹)

v₂ is the final velocity of object 2 (m s⁻¹).

Momentum When Masses Combine

If two objects combine when they collide, the equation is.

∑p_before = ∑p_after

∑mv_before = ∑mv_after

m₁u₁ + m₂u₂ = m₃v₃

m₁ is the mass of object 1 (kg)

u₁ is the initial velocity of object 1 (m s⁻¹)

m₂ is the mass of object 2 (kg)

u₂ is the initial velocity of object 2 (m s⁻¹)

m₃ is the combined mass of m1 and m2 (kg)

v₃ is the final velocity of combined mass of m₁ and m₂ (m s⁻¹).

Momentum in Explosive Collisions

∑mv_before = ∑mv_after

m₁u₁ = m₂v₂ + m₃v₃

m₁ is the mass of object 1 (2 and 3 combined; kg)

u₁ is the initial velocity of object 1 (m s⁻¹)

m₂ is the mass of object 2 (kg)

v₂ is the final velocity of object 2 (m s⁻¹)

m₃ is the mass of object 3 (kg)

v₃ is the final velocity of object 3 (m s⁻¹).

8.2 Change in Momentum and Impulse

• Increase in velocity = increase in momentum

• Decrease in velocity = decrease in momentum

• Change in momentum (Δp), is also called impulse (I)

p

Change in Momentum in One Dimension

• Impulse means change in momentum

I = Δp

= p_final − p_initial

= mv − mu

I is the impulse (kg m s⁻¹)

Δp is the change in momentum (kg m s⁻¹)

m is the mass (kg)

v is the final velocity (m s⁻¹)

u is the initial velocity (m s⁻¹).

Change in Momentum in Two Dimensions

I = mv − mu

= m(v − u)

8.3 Newton’s First Law

Force

• Force - push or pull, but exists in a variety of situations

    Measured in Newtons

    Vector quantity

• Contact Forces - forces that act directly on a body (Friction + drag force)

• Non-Contact Forces - forces that act on a body at a distance (Gravitation, magnetic and electric forces)

• If more than one force acts on a body at a time, the body behaves as if only one force - vector sum of all forces - is acting. Called Net force.

• The net force acting on a body experiencing a number of forces acting

  simultaneously is given by the vector sum of all the individual forces:

  F_net = F₁ + F₂ + … + F_n/

Newton’s First Law

• First law of inertia (the tendency of an object to maintain its velocity - related to the mass of an object)

Newton’s First Law → An object will maintain a constant velocity unless an unbalanced, external force acts on it

Therefore, an object will not maintain its velocity if an unbalanced, external force is applied.

‘External’, in relation to forces, implies that the forces are not internal. (When internal they have no effect on the motion)

Inertia

Inertia - the resistance to change in motion of an object

As the mass increases, the inertia increases and therefore;

• becomes harder to start moving of stationary

• becomes harder to stop moving

• becomes harder to change its direction of motion

Inertia effect is independent of gravity - since inertia depends on mass and weight force due to gravity also depends on mass

Newton’s First Law and Inertia

• Due to inertia, an object will continue with its motion unless a net force acts on the object

8.4 Newton’s Second Law

Newton’s Second Law - The acceleration of an object is directly proportional to the net force on the object and inversely proportional to the mass of the object:

F_net = ma

a is the acceleration of an object (in m s⁻²)

F_net is the force applied to the object (in N)

m is the mass of the object (in kg).

Can also be;

= m(v - u) / Δt

= (mv - mu) / Δt

= Δp / Δt

8.5 Newton’s Third Law

Newton’s Third Law - For every action (force), there is an equal and opposite reaction (force)

• when object A exerts a force F on object B, B will exert an equal and opposite force on A.

• Applies to contact and non-contact forces

Newton’s Third Law and Motion

• Is needed to explain all motion

Shown in this example, a component of the force acts down and another pushes backwards horizontally along the surface of the ground.

• Force is transmitted because there is friction between the shoe and the ground

• In response, the ground then pushes you forward via your foot

The Normal Force

• Action force is the force due to gravity of the Earth, Normal force is opposite that.

The Inclined Plane

8.6 Impulse and Force

Change in Momentum (Impulse)

• According to the 2nd law, a net force will cause a mass to accelerate

• Larger net force will create a faster change in the velocity of the mass

• The faster that occurs, the greater the net force that produced that change

F_net = Δp / Δt

        = m (v−u) / Δt

F_netΔt = m(v − u)

            = I

 I is the impulse (kg m s⁻¹).

Determining Impulse from a Changing Force

Is not always the case that the force that acted to change impulse over a period of time was constant

Example → tennis ball hit with racquet

• Instant ball comes into contact, applied force is small

• As strings distort and the ball compresses, force will increase until the ball has been stopped

• Force will then decrease as the ball accelerates away from the racquet

• The impulse will be I = F_avΔt

  • I is impulse

  • F_av is average force applied during collision

  • Δt is total period of time

Concept of impulse is appropriate when dealing with forces during any collision since it links force and contact time

8.7 Mass and Weight

Weight is vector

Mass is scalar

Mass of a Body

• Mass is measured in Kilograms (kg)

• Kilogram is defined in terms of an amount of a standard material

• An objects mass can be seen as a property by the amount of acceleration it undergoes for a given net force.

• More mass an object has, the greater the force is required to make it accelerate

Gravitational Force

• The force due to gravity is an attractive force that exists between all masses

Result from a mass creating a gravitational field that spreads throughout the space around the mass

• Any other mass that is within this field will experience a force towards the mass creating this field.

Weight Force

Weight - the force on a body due to gravity (F_s) or (W)

• Vector quantity

• Measures in Newtons

• Represented by a straight down arrow

The weight of a body F_g (in N) is defined as the force of attraction on a body due

to gravity and is calculated using the equation:

Fg = mg

F_g is the force of - gravity acting at the centre of mass of a body (in N)

m is the mass of the body (in kg)

g is the gravitational field strength (in N kg⁻¹, which is 9.8 N kg⁻¹ near the

surface of the Earth).

Your Mass and Weight on the Moon

• In space the matter that makes up your body doesn’t change

• Mass doesn’t change, because mass is a property of matter and is not affected by the environment

• Weight force will be much less on the moon than Earth

• Moons mass is 81 times smaller than Earth