Topic 5: Motion and Forces

Motion and Forces Overview

  • Motion in a straight line

    • Distance, speed, velocity, acceleration

    • Scalars and vectors

    • Forces and components

    • Newton's laws

    • Momentum, work, power, energy

  • Motion in a circle

  • Gravitation

  • Satellites

Scalars and Vectors

  • A scalar just tells us about the size (or magnitude) of the quantity.

  • A vector quantity has both size and direction associated with it.

  • Example:

    • Distance (e.g. 5m, 100km, 2mm) is a scalar quantity

    • Displacement (e.g. 5m S, 100km NE) is a vector quantity.

Speed and Velocity

  • Average (or constant) speed = distance travelled / time taken

  • Speed is a scalar quantity – size only

  • Average velocity = displacement / time taken

  • Velocity is a vector quantity – size and direction

  • The SI unit of speed and velocity is metres per second (m s- 1)

Velocity at an Instant in Time

  • Instantaneous velocity – use calculus approach

  • Consider a very short time interval Δt\Delta t

  • The displacement (vector distance) in time Δt\Delta t is Δs\Delta s

  • Velocity v=ΔsΔtv = \frac{\Delta s}{\Delta t} for smaller Δt\Delta t

  • limitting value Tend to use d rather than Δ\Delta

  • This is the “derivative of s” with respect to t →dsdt\rightarrow \frac{ds}{dt} or that we “differentiate s with respect to t”

Acceleration

  • Acceleration causes the velocity to change

  • Average acceleration = change in velocity / time taken for change

  • Acceleration at an instant =dvdt= \frac{dv}{dt}

  • Units of acceleration are ms−2m s^{-2}.

  • Forces produce accelerations. They change the speed or direction, or both.

Linear Motion with Uniform Acceleration

  • 3 formulas to apply here

    • v=u+atv = u + at

    • s=ut+12at2s = ut + \frac{1}{2} a t^2

    • v2=u2+2asv^2 = u^2 + 2as

    • Where:

      • u = initial velocity

      • v = final velocity

      • a = acceleration

      • s = distance travelled (displacement)

      • t = time

Example 1

  • What is the acceleration of a car that increases its velocity from 20ms- 1 to 60ms- 1 in 4 seconds?

    • u = 20ms- 1

    • v = 60ms- 1

    • t = 4s

    • a = ?

    • Use v=u+atv = u + at

    • a=(v–u)t=(60–20)4=10ms−2a = \frac{(v – u)}{ t} = \frac{(60 – 20)}{ 4} = 10 ms^{-2}

Example 2

  • You drop a stone down a well and hear it hit the bottom after 2 seconds. What is the depth of the well? Take g=10ms−2g = 10ms^{-2}.

    • u = 0ms- 1

    • a = g = 10ms- 2

    • t = 2s

    • s = ?

    • Use s=ut+12at2=(0×2)+(12×10×22)=20ms = ut + \frac{1}{2} a t^2 = (0 \times 2) + (\frac{1}{2} \times 10 \times 2^2 ) = 20m

Graphical Approach

  • Plot distance / time graph and gradient gives velocity at any time.

  • Plot velocity / time graph and gradient gives acceleration at any time.

  • Area under the curve gives distance travelled.

Adding Vectors

  • A boat is crossing from one side of a river to the other. The boat’s engine produces a velocity of 5ms- 1. The river current travels at 12ms- 1. What is the resultant velocity of the boat?

Resolving Components of a Vector

  • A vector can be resolved into two perpendicular components:

  • Useful to enable to add vectors together.

Example of Resolving Forces

  • A body moving down a slope under gravity (e.g. avalanche, mud slide):

  • A rock of mass 100 kg is moving down a 10° slope. If it starts from rest, how long will it take to travel down a 500 m slope?

    • Acceleration down slope = component of force down slope / mass of rock

      • =(100kg×10ms−2×sin10°)100kg=1.736ms−2= \frac{(100 kg \times 10 ms^{-2} \times sin 10°)}{ 100 kg} = 1.736 ms^{-2}

    • Use s=ut+12at2s = ut + \frac{1}{2} a t^2

      • 500m=0+(12×1.736ms−2×t2)500 m = 0 + (\frac{1}{2} \times 1.736 ms^{-2} \times t^2)

      • gives t = 24.0 s

Newton’s Laws of Motion

  • 1st Law: A body continues in a state of rest or uniform motion unless acted upon by a force.

  • 2nd Law: Force = mass x acceleration

    • where:

      • F = force in N (Newtons)

      • m = mass in kg

      • a = acceleration in ms- 2

    • F=maF = ma

  • 3rd Law: When two bodies interact each exerts an equal and opposite force on the other.

Example of Resolving Forces

  • Pulling a barge along a canal. A horse on the bank tows the barge along with a force, F, of 500N. The tow rope makes an angle of 42° with the direction of travel of the barge. If the barge has a mass of 1500 kg, what is its acceleration?

    • Acceleration = component of force along river / mass of barge

      • =500N×cos42°1500kg=371.57N1500kg=0.2477ms−2=0.25ms−2= \frac{500 N \times cos 42°}{ 1500 kg} = \frac{371.57 N}{ 1500 kg} = 0.2477 ms^{-2} = 0.25 ms^{-2}

      • to 2 s.f.

Mass and Weight

  • Mass, m, is the amount of substance (in kg) in a body.

  • Weight, W, is the force of gravity (in N) acting on the body towards the centre of the Earth.

    • W=mgW = mg

Example 3

  • A rocket with lift-off mass of 2.8×106kg2.8 \times 10^6 kg develops an initial thrust of 3.3×107N3.3 \times 10^7 N. What is the initial acceleration on lift-off? Take g=10ms−2g = 10 ms^{-2}.

    • Resultant upward force F on the rocket = T – W.

    • F=masoT–W=maF = ma so T – W = ma

    • a=(T−W)m=(3.3×107–2.8×106×10)(2.8×106)=1.8ms−2a = \frac{(T-W)}{ m} = \frac{(3.3 \times 10^7 – 2.8 \times 10^6 \times 10)}{ (2.8 \times 10^6)} = 1.8 ms^{-2}

Friction

  • The frictional force between two surfaces always opposes their relative motion.

  • Frictional force depends on:

    • The nature of the surfaces in contact

    • The normal reaction, N

  • Frictional force does not depend on the area in contact.

  • Frictional force F=μNF = \mu N

    • where μ\mu is the coefficient of friction.

Momentum

  • Momentum = mass x velocity

    • p=mvp = mv

Momentum

  • Momentum = mass x velocity

    • p=mvp = mv

  • Principle of conservation of momentum: When bodies interact, the total momentum remains constant (provided no external force acts on the system).

  • Momentum is conserved.

Rate of Change of Momentum

  • p=mvp = mv (p = momentum)

  • Rate of change of momentum is equal to the applied force.

  • This is another form of Newton’s 2nd Law.

How Many Forces Do We Know?

  • Fundamental Forces

    • Strong Nuclear Force: Holds nucleus together, Strength 1, Range 10-15 m

    • Electromagnetic Force: Strength 1/137, Infinite Range

    • Weak Nuclear Force: Induces beta decay, Strength 10-6, Range 10-18

    • Gravity: Strength 6 x 10-39, Infinite Range

Energy and Work

  • Energy can be thought of as a measure of “a body’s capacity to do work”.

  • Work is done when “the point of application of a force is moved along line of action of the force” (i.e. component of the force in that direction).

  • Work done requires to consider also supply or loss of heat (as well as D in internal energy) (see 1st law of Thermodynamics, lectures 5-7)

Work

  • Work done = Force x distance moved (with force and displacement in the same direction)

  • But Work and energy are scalar quantities, whereas force is a vector quantity.

  • Use vector multiplication, work done W=F.dW = F . d

    • (“dot product” à component of displacement in direction of F)

  • . = dot product [d = vector displacement] [a kind-of vector multiplication]

  • A force which is at 90°to the motion does no work (F . d = 0 in this case)

Power

  • Power is the rate of doing work

  • Power = work done / time taken

    • = energy supplied or consumed / time

Example 4

  • A cyclist maintains her speed at 7 ms- 1 (against friction / air resistance) given at this speed force = 30 N What is her average power output?

    • Power = work done / time

      • = (force x distance moved) / time

    • Consider a time of 1 second.

    • Power = (30 N x 7 ms- 1 x 1s) / 1s

      • = 210 W

Power

  • Power is the rate of doing work

  • Power = work done / time taken

    • = energy supplied or consumed / time

  • Power = force x velocity (cpt in direction of force)

  • Again, use dot product to “multiply” vectors P=F.vP = F . v

  • Units of energy and work are Joules (J)

  • Units of power are Watts (W)

  • 1 W = 1 J s- 1

Mechanical Energy

  • Two forms of mechanical energy:

    • Kinetic energy (due to motion)

    • Potential energy (stored energy)

  • Kinetic energy depends on mass and velocity

    • KE=12mv2KE = \frac{1}{2} m v^2

Gravitational Potential Energy

  • This is energy a body has because of its position in a gravitational field.

  • The amount of gravitational PE depends on

    • mass, m

    • height, h

    • acceleration due to gravity, g

  • A falling body loses PE but gains KE.

    • PE=mghPE = mgh

Hydroelectric Power Generation

  • The interchange of PE and KE is the basis of Hydroelectric power generation.

  • Typically:

    • Water falls from a high mountain reservoir to a HEP station below.

    • As it falls, it loses PE and gains KE.

    • The KE of the falling water drives a generator at the power station.

  • Maximum power requires

    • Large height difference

    • Large mass of water per second

HEP Pumped Storage

  • When demand for electricity is low: Surplus energy is used to pump water up to a high reservoir.

  • When demand for electricity is high: Water is allowed to fall down from the reservoir to increase electricity generation.

  • Smoothes out some of the variability in electricity demand.

Motion in a Circle

  • Speed is constant Direction changes

Circular Motion – The Basics

  • Distance around a circle is measured as:

    • An angle θ (in radians)

    • A length on the circumference

      • s (in m)

  • Velocity around a circle is measured as:

    • Linear velocity

      • v (in ms- 1)

    • Angular velocity

      • ω (in rad s- 1)

    • s=rθs = r\theta

    • v=rωv = r\omega

    • n.b. circumference = 2π\pi r à measure angle in radians, with 360o = 2π\pi radians

Angular Velocity

  • On a disc or wheel:

    • Linear velocity, v, increases with r

    • Angular velocity, ω, stays constant as r increases.

  • Differentiating this with respect to time gives:

Angular Velocity and the Earth

  • Compare the motion of a point on the equator with a point at 52°N as the Earth rotates.

    • Each point takes 24 hours to complete one revolution.

    • The angular velocity, ω, of both points is the same

    • The point at 52°N is travelling in a circle of smaller radius than the point on the equator.

    • The linear velocity, v, of the point at the equator is faster.

Motion in a Circle

  • Centripetal force acts towards the centre of the circle producing a centripetal acceleration.

Centripetal Acceleration and Centripetal Force

  • Centripetal acceleration a=rω2=v2ra = r\omega^2 = \frac{v^2}{ r}

  • Centripetal force F=ma=mrω2=mv2rF = ma = mr\omega^2 = \frac{mv^2}{ r}

  • The direction of a and F are both towards the centre of the circle.

  • If an object on a string is swung in a circle, F is provided by the tension in the string.

  • For satellites, planets, stars, etc, F is provided by the gravitational force between the bodies

Gravitation

  • Bodies exert a gravitational force on each other because of the matter they contain.

  • Gravitational forces:

    • Maintain the position of the Sun, Moon, stars, planets.

    • Ensure that the Earth retains its atmosphere

    • Hold us (and everything else) on the surface of the Earth.

  • Size of the gravitational force depends on:

    • Masses of the bodies

    • Separation of the bodies

Gravitational Force

  • Gravitational force, F, between two bodies with masses M1M_{1}and M2M_2 , and separated by a distance r, is given by

  • F=GM1M2r2F=\frac{GM_1M_2}{r^2}

    • Where G is the universal Gravitational constant and has the value 6.7×10−11Nm2kg−26.7 \times 10^{-11} N m^2 kg^{-2}.

    • Note:

      • (1) G is tiny – F is difficult to measure unless the masses are large

      • (2) Gravitational force follows an inverse square law

Gravitational Force on Earth

  • For a person, mass mpm_{p} , standing on the Earth, mass MpM_{p}:

  • F=GMEmpREF=\frac{GM_{E}m_{p}}{R_{E}}

    • where RER_E is the radius of the Earth.

  • Comparing this with F=mgF = mg gives:

  • g=GMEREg=\frac{GM_{E}}{R_{E}}

Is g Constant Over the Earth?

  • g depends on the radius of the Earth, RER_E

  • Earth is not a perfect sphere - RER_E varies

  • g can vary from 9.78 to 9.82 ms- 2 over the surface of the Earth

  • Weight varies by about 0.1% (1 g per kg)

  • Where would you go on the Earth to weigh the least?

The GRACE* Satellite Mission

  • *Gravity Recovery and Climate Experiment

  • Microwave ranging measures distance between two polar- orbiting satellites to detect variations in the Earth’s gravitational field from very slight changes in the orbit of 1st satellite relative to 2nd

The GRACE Satellite Mission

  • This gravity map from the GRACE satellite shows (from the perspective of the satellite) the variation of the gravitational field across Earth’s surface; red indicates stronger gravitational field (e.g. higher altitude) and blue indicates weaker gravity (~01% anomaly relative to).

Satellites

  • The gravitational force supplies the centripetal force which keeps a satellite (mass msm_{s}, velocity vsv_{s}) in orbit:

  • F=GMEmsR2=msvs2RF=\frac{GM_{E}m_{s}}{R^2}=\frac{m_{s}v_{s}^2}{R}

  • For a given mass of satellite, the radius R of “geostationary” orbit can be calculated (rotates at same angular velocity)

  • R is the radius of the orbit of the satellite

Kinetic Energy of a Rotating Body

  • Every “particle” of mass mpm_{p} within a rotating body has a (KE)p(KE)_{p}.

    • (e.g. think of juggling hammers [in motion])

  • (KE)p=12mpv2=12mp(rω)2=12mprp2ω2\left(KE\right)_{p}=\frac12m_{p}v^2=\frac12m_{p}\left(r\omega\right)^2=\frac12m_{p}r_{p}^2\omega^2

  • Total KE of complex body = Σ12mprp2ω2=12ω2Σmprp2\Sigma\frac12m_{p}r_{p}^2\omega^2=\frac12\omega^2\Sigma m_{p}r_{p}^2

  • Σmprp2\Sigma m_{p}r_{p}^2 is the moment of inertia, I.

  • Total (angular) KE of body = 12Iω2\frac{1}{2} I \omega^2

Angular Momentum

  • Linear momentum depends on mass and velocity, p=mvp = mv

  • Angular momentum depends on the distribution of the mass with respect to the axis of rotation I, and the angular velocity ω.

  • Angular momentum = IωI\omega

  • Total angular momentum of a system is conserved

    • (e.g. ice skater spins round faster when arms are closer to the body).

Conservation of Angular Momentum in the Atmosphere

  • Consider a rotating turbulent cloud.

    • Vertical motion stretches the cloud in the vertical and squashes it in the horizontal

    • More of the cloud is closer to the axis of rotation.

      • I decreases.

    • To conserve angular momentum, w increases.

    • Cloud rotates faster.

Dimensional Analysis

  • In lecture 1 we talked about checking units agreed to check for mistakes in an equation.

  • More formally leads to the idea of dimensional analysis.

  • Most quantities we encounter can be written in terms of kg, m and s.

Dimensional Analysis

  • Irrespective of the units (mph, ms- 1) a velocity is always a length / time.

    • i.e. v=[L][T]−1v = [L] [T]^{-1}

  • Force (= ma) is always mass x length / time2

    • i.e. F=[M][L][T]−2F = [M] [L] [T] ^{-2}

  • The letters in square brackets are dimensions

  • Most quantities can be expressed as combination of [M], [L] and [T].

Dimensional Analysis More Examples:

  • Work = force x distance = [M][L][T]−2[L]=[M][L]2[T]−2[M] [L] [T] ^{-2} [L] = [M] [L]^2 [T] ^{-2}

  • Pressure = force / area = [M][L][T]−2/[L]2=[M][L]−1[T]−2[M] [L] [T] ^{-2} / [L]^2 = [M] [L] ^{-1} [T] ^{-2}

  • If an equation is correct, the dimensions on both sides are the same – a useful check.

  • Can also be used to deduce the relationship between different variables.

Summary

  • Motion in a straight line

    • Distance, speed, velocity, acceleration

    • Scalars and vectors

    • Forces and components

    • Newton's laws

    • Momentum, work, power, energy

  • Motion in a circle

  • Gravitation

  • Satellites

  • Dimensional analysis