Dr Chris Skilbeck Newton’s Laws of Motion An Introduction to Mechanics
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Dr Chris Skilbeck
Newton’s Laws of Motion
An Introduction to Mechanics
Topics Covered:
1. Newton’s Laws of Motion
2. Force Diagrams
3. Introduction to Calculus and vectors
The Laws of Motion
Aristotle (384-322 BC)1. The natural state of motion is
rest
2. Where there is motion, it must be caused by an external agent or force that overcomes the natural reluctance of objects to move
Why are these laws incorrect?!
The Laws of MotionIsaac Newton (1642 – 1727)1. The natural state of motion is
uniform progress along a straight line – motion is constant at constant velocity and zero acceleration in the absence of forces
2. Acceleration is caused by external forces
Newton’s Laws of Motion
1. An object either moves in a straight line at constant velocity or remains at rest if no external force acts upon it
2. Force = mass x acceleration: F = ma
Force and acceleration are vector quantities, mass is a scalar quantity
3. For every force there is an equal and opposite reaction
F = ma a = F/mx-axis
Motion in One Dimension
m = 50 g, a = 9.81 m/s2
a is acceleration due to gravity
Q.1 What is F?
Vectors
• Force and acceleration are vectors
• They have both magnitude and direction
• Can be represented in diagrams by arrows
• Length of arrow = magnitude
• Vectors written as: a or a or ā
• Magnitude of a = |a| = a
• Vectors have components, e.g. F = Fx + Fy + Fz
kFjFiFF zyxˆˆˆ
kji ˆ,ˆ,ˆ Are unit vectors: point in positive x, y and z-axis directions, but have magnitude = 1
z
x
y
Coordinate Systems
ij
k
Left or Right-handed coordinates?
Resolving Vectors
y
x
r (7,5)
What are the lengths of the components of the vector r, which lies in the y and x-directions (ryand rx respectively)?
Hint: SOHCAHTOA
Radians
Circle: radius r
circumference = 2r
area = r2
360o or 2 rad
r
r arc length, L
L = r
with in rad
1 rad ≈ 57o 17' 45''
When L = r:
= 1 rad
So: 1 rad is the angle of the sector when L = r
Area of sector: A = ½r2
o180radiansin angle degreesin Angle
Q.2 How many radians is 180o?
Q.3 How many degrees is 2/ rad?
Q.4 how many radians is 120 degrees?
Motion in One Dimension
• We can simplify by choosing the x-axis to be parallel to direction of motion
• Vector equations reduce to scalar equations: a force acting towards +x is positive, a force acting towards –x is negative
F = ma a = F/mx-axis
Motion in One Dimension
m = 50 g, a = 9.81 m/s2
a is acceleration due to gravity
Q.1 What is F?
Constant Acceleration in One Dimension
• Close to the surface of the Earth the force of gravity is approximately constant = 9.81 m/s2 at surface of earth
• Acceleration due to gravity is approximately constant: a = c
vx
t
vx = at + ux
ax
t
ax = 9.81 m/s2
9.81
Uniformly accelerated motion
uxQ. 5 What is ux?
Practice Questions
Q. 6
A sky-jumper is in free-fall. What is their velocity after 10 seconds? [g = 9.81 m/s2; assume there is no air resistance.]
Q.7
In practice, the terminal ‘velocity’ of a skydiver is about 55 m/s. Why is this?
Introduction to Calculus
• Calculus studies rates of change / how one variable changes as another changes, e.g. how velocity changes over time
Acceleration = change in velocity / change in time
tva
This is the gradient of the velocity-time curve!
Acceleration is a function of time: a = a(t)
Introduction to Calculus
Q
txv
Velocity = change in distance / change in time
This is the gradient of the distance-time curve!
x
t
distance, x
time, t
x + x
x
t + tt0
How do we get the gradient at point
Q = (x + ½x, t + ½t)
?
Qx
t
distance, x(t)
time, t
x(t + t)
x(t)
t + tt0
Introduction to Calculus
1D motion: distance moved along the x-axis is a function of time: x = x(t)
gradient =x(t + t) – x(t)
t + t - t
x(t + t) – x(t)
t=
Qx
t
distance, x(t)
time, t
x(t + t)
x(t)
t + tt0
Introduction to Calculus
1D motion: distance moved along the x-axis is a function of time: x = x(t)
gradient at Q =x(t + t) – x(t)
tlim
t → 0
x
tlim
t → 0=
=dtdx
Introduction to Calculus - Limits
• Mathematics is not always intuitive!
• What is the sum of the following series for an infinite number of terms?
18 + 1.8 +0.18 +0.018 +0.0018 + 0.00018 + …
S2 = 19.8
S3 = 19.98
S4 = 19.998
S5 = 19.9998
S6 = 19.99998
S7 = 19.999998
For an infinite number of terms the sum S∞ = 20
As we can never count to infinity we say: Sn → ∞ as n →∞
or lim Sn = 20
The limit of Sn tends to 20 as n tends to infinity
n→∞
Introduction to Calculus - Differentiation
• Differentiation gives us the gradient of a curve at a point
• This is immensely useful !!!!!
From the distance-time graph, the velocity at any instant of time is the gradient at time t:
v = dx / dt
From the velocity-time graph, the acceleration at any instant of time is the gradient at time t:
a = dv / dt
Introduction to Calculus - Differentiation
For a function f(t), the derivative of f with respects to t, df/dt:
f(t) = c = constant df/dt = 0 why?
f(t) = t df/dt = 1 why?
f(t) = 2t df/dt = 2 why?
f(t) = t2 df/dt = 2t
General:
f(t) = htn + g df/dt = nhtn-1
Introduction to Calculus - Differentiation
For y = y(x), where a, b = constant
acos(ax)sin(ax)
-asin(ax)cos(ax)
aeaxeax = exp(ax)
1/x (for ax > 0)ln(ax) = loge(ax)
abxa-1bxa
axa-1xa
0a
derivative dy/dxFunction y(x)
• v = v(t) for motion along the x-axis we can use scalar quantities, velocity is a vector, speed is a scalar, so: v = v(t)
• a = a(t)
• v = dx/dt
• a = dv/dt
For uniformly accelerated motion:
|a| = c = constant
How do we obtain v and x ?
Introduction to Calculus - Differentiation
Q.8 A particle of mass 2 kg oscillates along the x-axis according to the equation:
65sin2.0 tx
Where x is in m, t in s.
(a)What is the force acting on the particle at t = 0?
(b)What is the maximum force that acts on the particle?
Introduction to Calculus - Integration
• Integration is the reverse of differentiation!
• Differentiation gives us the gradient of a curve
• Integration gives us the area under a curve
• dx/dt is the derivative of x with respects to t
• ∫x(t)dt is the integral of x with respects to t
y
x
y = x
dy/dx = 1
The gradient = 1 for all points along the line!
Area under the curve
dy/dx of (y = xa) = axa-1
aa xdxax 1
21
21 xdxxxdx
11 aa xa
dxx
1 aa axxdxd
aa xxadx
d11
Introduction to Calculus - Integration
Area of a triangle?
Introduction to Calculus - IntegrationFor y = y(x), where a, b = constant
cos(ax)
sin(ax)
eax
xa for a ≠ -1axa
a = constant0
integral ∫y(x)dxFunction y(x)
1
1
axa
axea1
)cos(1 axa
)sin(1 axa
Equations of Motion
For uniformly accelerated motion in one direction: adtdv
Integrate w.r.t. t to get v: catv
What is the constant C? It is the initial velocity, u (or v0), of the object at time t = 0, when v = u, so:
uatv
Integrate w.r.t. t to get distance moved, s:
utats 2
21
Equations of Motion – Constant Acceleration Equations
Uniformly accelerated motion along x-axis: a = constant
constant xx a
dtdv
xxx utav
tutas xxx 2
21
Fx = max
Uniform Motion and Newton’s First Law
An object either moves in a straight line at constant velocity or remains at rest if no external force acts upon it
Uniform motion: No force acts on the object, its acceleration is zero
In 1D:
ax = 0
vx = ux
sx = uxt
The uniform motion equations
Practice Questions
Q.9
A spaceship is drifting in space, moving in a straight line with speed 12 km/s. How far will it have drifted: a) in 1 hour, b) in one year? [Assume negligible ‘air’resistance.]
Q.10
A man drops from a 30 metre balcony. Assuming he was initially stationary, how long will it take him to reach the ground? What will be his speed on impact? [g = 9.81 m/s2].
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