Download The main difference between scalars and

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▪ Magnitude – A numerical value with appropriate units.
▪ Scalar is a quantity that is completely specified by magnitude.
▪ Vector requires both, magnitude and direction for a complete description.
The main difference between scalars and vectors is difference between scalar and vector algebra
 If you know the magnitude A and direction θ of a vector A = (A, θ) you can find x and y components of that vector:
v = 34 m/s @ 48° . Find vx and vy
A x = Acosθ
if (A,θ) 
A y = Asinθ
known
vx = (34 m/s )cos 48 ) = 23 m/s;
vy = (34 m/s) sin 48° = 25 m/s
 If you know x and y components of a vector A you can find the magnitude A and direction θ of that vector:
A= A2x +A2y
if (A x ,A y ) 
known
A 
θ=arc tan  y 
 Ax 
Fx = 4 N and Fy = 3 N .
if the vector is in the first quadrant;
if not you find it from the picture.
F = 42 +32 = 5N ;
 = arc tan (¾) = 370
The sum is the vector sum of the two individual vectors, known as the "resultant" or “net vector
"
 
 SUBTRACTION is adding opposite vector.
C = A - B = A + -B
▪ ADDITION ANALYTICALLY/NUMERICALLY:
EXAMPLE:
F1 = 68 N @ 24°
F2 = 32 N @ 65°
1. Draw a sketch
2. Find components of the resultant vector
Find
F  F1  F2
Fx = F1x + F2x = 68 cos240 + 32 cos650 = 75.6 N
Fy = F1y + F2y = 68 sin240 + 32 sin650 = 56.7 N
3. Knowing components of the resultant vector, find its magnitude and direction:
F  Fx2  Fy2  94.5 N
 = arc tan (56.7/75.6) = 36.90
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● Displacement is an object’s change in position
∆x = x2 – x1
where x2 is the final position and x1 is the initial position
● Average velocity is the displacement covered in unit time
∆𝑥
𝑣𝑎𝑣𝑔 =
𝑣 ℎ𝑎𝑠 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 − 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑎𝑠 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
𝑡
● Average and Instantaneous velocity
Instantaneous velocity is the velocity at one instant. There is no formula for it using algebra.
Only if one knows x(t), then derivative of it would give v(t).
● Average speed is the distance covered in unit time
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑣𝑎𝑣𝑔 =
𝑖𝑡 𝑡𝑒𝑙𝑙𝑠 𝑢𝑠 ℎ𝑜𝑤 𝑓𝑎𝑠𝑡 𝑡ℎ𝑒 𝑜𝑏𝑗𝑒𝑐𝑡 𝑖𝑠 𝑚𝑜𝑣𝑖𝑛𝑔 𝑟𝑒𝑎𝑙𝑙𝑦
𝑡
Unfortunately we use the same letter for velocity and speed: v. Velocity should be: 𝑣⃗. Speed should be:v
● Acceleration is change in velocity per unit time
𝑎=
∆𝑣
∆𝑡
 Any Motion
𝑓𝑟𝑜𝑚 𝑑𝑒𝑓𝑖𝑡𝑖𝑜𝑛: 𝑣𝑎𝑣𝑔 =
𝑠
→ 𝑠 = 𝑣𝑎𝑣𝑔 𝑡
𝑡
 Motion with constant velocity – uniform motion
v = vavg at all times, therefore:
s = vt
 Uniform Accelerated Motion equations
𝑠 = 𝑣𝑎𝑣𝑔 𝑡 𝑎𝑙𝑤𝑎𝑦𝑠 − 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑚𝑜𝑡𝑖𝑜𝑛
𝑣 = 𝑢 + 𝑎𝑡
𝑢 + 𝑣
They can only be used if the
𝑣𝑎𝑣𝑔 =
2
acceleration a is CONSTANT.
𝑎 2
𝑠 = 𝑢𝑡 + 𝑡
2
2
2
𝑣 = 𝑢 + 2𝑎𝑠
● Free fall is vertical (up and/or down) motion of a body where gravitational force is the only
force acting upon it. (when air resistance can be ignored).
Gravitational force gives all bodies regardless of mass or shape, the same acceleration when air resistance can be ignored.
For an object in free fall the speed would decrease by 10 m/s every second on the way up, at the top it would reach zero,
and increase by 10 m/s for each successive second on the way down
Formulas for uniform accelerated motion with a=g=10 m/s2, downward
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● ALL WE KNOW ABOUT GRAPHS
Velocity at some time is
equal to the slope of the
tangent line at that
position on the positiontime graph.
Acceleration at some
time is equal to the slope
of the tangent line at
that position on the
velocity-time graph.
Area under the
velocity-time graph
between two
times/positions is the
displacement covered
in that time interval.
● Air resistance provides a drag force to objects in free fall.
Air resistance is speed dependent. The drag force increases as the speed
of the falling object increases resulting in decreasing downward
acceleration. When the drag force reaches the magnitude of the
gravitational force, the falling object will stop accelerating and continue
falling at a constant velocity. This is called the terminal velocity/speed.
Average velocity
between two positions is
equal to the slope of the
secant line between
these two points on the
position-time graph.
Average acceleration
between two positions is
equal to the slope of the
secant line between
these two points on the
velocity-time graph.
Area under the
acceleration-time
graph between two
times/positions is the
change in velocity in
that time interval.
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● Projectile motion
- motion of objects (thrown or projected into the air with an initial velocity
gravitational force if we can neglect air resistance.
u ) upon which the only force is
Projectile motion is combined motion of the two independent motions simultaneously
● one in horizontal (x) direction with constant velocity vx = ux (no force in x direction  no acceleration
 no change in velocity in that direction)
● one in vertical (y) direction with constant acceleration g downwards (vertical component of velocity
changes as in the free fall)
● velocity has the smallest magnitude at maximum height, because vertical component of velocity is zero there
HORIZONTAL MOTION
ux = u cos 0
vx = ux
VERTICAL MOTION
uy = u sin 0
vy = uy + gt
v2y =u2y + 2gy
g 2
t
2
u  vy
y= y
t
2
y = uy t +
x = ux t
● Inertia is resistance an object has to a change of velocity
● Mass is numerical measure of the inertia of a body (kg)
● Weight is the gravitational force acting on an object . W = mg
● Force is an influence on an object that causes the object to accelerate.
● 1 N is the force that causes a 1-kg object to accelerate 1 m/s2
● ∑ ⃗⃗⃗⃗
𝑭 (⃗⃗⃗⃗
𝑭 𝒏𝒆𝒕 ) (resultant/net force) is the vector sum of all forces acting on an object
●
Free Body Diagram is a sketch of a body (point object) and all forces acting on it
● Newton’s first law: An object at rest stays at rest and an object in motion stays in motion
with the same speed and in the same direction unless acted upon
by a net external force.
Translational equilibrium: ∑ 𝐹⃗ = 0 ⇔ 𝑎⃗ = 0 ⇒ ∆𝑣⃗ = 0
 if net F = 0 then a = 0, and velocity is constant or zero
 if velocity is constant or zero, then a = 0, and Fnet = 0
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● Newton’s second law:
∑ 𝐹⃗ =
The rate of change of momentum of an object is directly proportional to the
resultant force applied and is in the direction of the resultant force.
The resultant force is equal to the rate of change of momentum.
∆𝑝⃗
∆𝑡
If mass doesn’t change: Δp = mΔv → F = m
●
Newton’s third law:
∆𝑣
= ma
∆𝑡
Whenever object A exerts force on object B, object B exerts an equal in magnitude,
but opposite in direction force on object A.
We are talking about forces acting on two different bodies.
●
Normal force Fn is the force which is preventing an object from falling through the surface of another body.
That’s why normal force is always perpendicular (normal) to the surfaces in contact.
●
Tension T is a force that the end of the rope exerts on whatever is attached to it.
Direction of tension is along the rope.
●
Spring force is proportional to extension/compression: Fs = - k ∆x
Constant of proportionality is called spring constant.
The negative sign indicates that the spring force always tries to bring spring into equilibrium position.
●
Friction force is the force that opposes slipping (relative motion ) between two surfaces in contact
▪ It acts parallel to surface in direction opposed to slipping.
▪ Friction depends on type and roughness of surfaces and normal force.
▪ Magnitude of the friction force is:
Ff = μ Fn
μ is called coefficient of friction
• kinetic μ is smaller than static μ.
●
Uniform circular motion
Motion with constant speed around circle. Velocity vector is tangent to the path at each
instant, direction of velocity vector changes all the time as the object moves in circle.
Object undergoing uniform circular motion is accelerating with centripetal acceleration ac .
It has centripetal force acting upon it:
Fc = mac =
●
mv 2
r
Newton’s Law of Universal Gravitation
The force between two objects is proportional to the product of their masses
and inversely proportional to the square of the distance between their centers.
F=G
m1 m2
r2
G = 6.67x10-11 Nm2/ kg2 – “Universal gravitational constant”
Weight is force of gravity acting on object m.
F=G
Mm
R2
= mg
g=G
M
R2
for the Earth 9.80 m/s2
R – radius of the Earth is distance between Earth and m
M – mass of the Earth
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F sin θ + Fn=mg
Fn = mg cos θ
●
Work done by constant force is the product of the component of the force in the direction of
displacement and the magnitude of the displacement. (scalar)
W = Fd cos Ѳ
(Fd = F cos Ѳ)
(Joules)
●
Work done by a varying force F along the whole distance travelled
is the area under the graph FcosѲ
versus distance travelled.
●
Work done by external force F = kx when extending a spring from extension x1 to x2 is:𝑊 = 𝑘(𝑥22 − 𝑥12 )
●
Gravitational potential energy GPE = mgh
●
Elastic potential energy stored in spring when extendex/compressed by 𝑥 𝐸𝑃𝐸 = 𝑘𝑥 2
●
Kinetic energy is the energy an object possesses due to motion KE = ½ mv2
●
Work – Kinetic energy theorem: work done by net force changes kinetic energy:
W = ∆KE = final EK – initial EK = ½ mv2 – ½ mu2
1
2
1
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● Conservation of energy law: Energy cannot be created or destroyed. It can only be changed from one form to another.
For the system that has only mechanical energy (ME = potential energies + kinetic energy) and there is no frictional force
acting on it, so no mechanical energy is converted into thermal energy, mechanical energy is conserved
ME1 = ME2 = ME3 = ME4
mgh1 + ½ mv12 = mgh2 + ½ mv22 = • • • • • •
If friction cannot be neglected we have to take into account work done by friction force which doesn’t belong to the object
alone but is shared with environment as thermal energy. Friction converts part of kinetic energy of the object
into thermal energy. Frictional force has dissipated energy:
ME1 + Wfr = ME2
(Wfr = – Ffr d)
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●
or
Power is the rate at which
▪ work is performed
▪ energy is transmitted/converted.
𝑃=
𝑊
𝑡
𝑜𝑟
𝑃=
∆𝐸
𝑡
1𝑊 (𝑊𝑎𝑡𝑡) =
1𝐽
1𝑠
P = Fv cos 
●
Efficiency is the ratio of how much work, energy or power we get out of a system compared to how much is put in.
eff =
●
Wout
E
P
= out = out
Win
Ein
Pin
Linear momentum is defined as the product of an object’s mass and its velocity.
𝑝⃗ = 𝑚𝑣⃗
𝑣𝑒𝑐𝑡𝑜𝑟!
(𝑝) = kg m/s
●
Impulse due to constant force acting is defined as the product of the net force
acting on an object and time interval of action:
(𝐼) = 𝑁𝑠
𝐼⃗ = 𝐹⃗ ∆𝑡
𝑣𝑒𝑐𝑡𝑜𝑟!
●
Impulse done by a varying force F acting over time interval ∆𝑡 is
the area the area under
the force-time graph.
●
Impulse – momentum theorem states that the change in momentum of an object
equals the impulse applied to it.
F∆t = ∆p
Δp = mv - mu Ns = kg m/s
(logically equivalent to Newton's second law of motion)
●
Law of Conservation of Momentum: The total momentum of a system of interacting
particles is conserved (remains constant, provided
there is no resultant external force.
Such a system is called an “isolated system”.
𝑝⃗𝑠𝑦𝑠𝑡𝑒𝑚 = ∑ 𝑝⃗
𝑖
∆𝑝⃗𝑠𝑦𝑠𝑡𝑒𝑚 = 0
∴ momentum of the system after collision = momentum of the system before collision
REMEMBER TO DRAW A SKETCH OF THE MASSES AND VELOCITIES BEFORE AND AFTER COLLISION.
LABEL VELOCITIES AS POSITIVE OR NEGATIVE.
Elastic collision: both momentum and kinetic/mechanical energy are conserved.
That means no energy is converted into thermal energy
Inelastic collision: momentum is conserved but KE is not conserved.
Perfectly inelastic collision: the most of KE is converted into other forms of energy when objects after
collision stick together.
To find how much of KE is converted into thermal energy, subtract KE of the system after collision
from KE of the system before the collision.
If explosion happens in an isolated system momentum is conserved but KE increases