1. Vectors & Equilibrium
Sara sultan
07/04/2019

2. Motion in 1 Dimension
Motion is defined as change in position of a body over time with respect to the surrounding. For the concept of motion in one direction, consider a motion of a particle along a straight track. A B
Distance: Measurement of any length b/w point A & B.
Displacement: A change of position of a particle along a straight path (shortest distance).
Position:
Position, when restricted to the x-axis, can be described as a single number x.
If x(t)= 2t, x(3) = ? Position along axis? For x(t)= t2 + 5, find position at t=7.
If the particle does not remain in the same place at different times, then the value of x will change in time, and thus we can define a function x(t) which returns the position of the particle at a point in time t.
07/04/2019

3. Vectors (Directional quantities)
A vector has magnitude (size) and direction.
The length of the line shows its magnitude and arrowhead points in the direction.
We can add two vectors by joining them head-to-tail.
Commutative property holds in vectors.
And it doesn't matter which order we add them, we get the same result.
A vector is represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by AB.
The magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement from A to B.
The velocity and acceleration of a moving object and the forces acting on it can all be described with vectors.
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4. Speed & Velocity
Imagine that on your way to class one morning, you walk at 3 m/s east towards campus. After one minute you realize that you've left your assignment at home, so you turn around and run, at 6 m/s. You're running twice as fast as you walked, so it takes half as long (30 seconds) to get home again.
Calculate your average speed, which is defined as the total distance covered divided by the time. If you walked for 60 seconds at 3 m/s, you covered 180 m. You covered the same distance on the way back, so you went 360 m in 90 seconds.
Average speed = distance / elapsed time = 360 / 90 = 4 m/s.
Average velocity = displacement / elapsed time.
In this case, your average velocity for the round trip is zero, because you're back where you started so the displacement is zero.
We usually think about speed and velocity in terms of their instantaneous values, which tell us how fast, and, for velocity, in what direction an object is traveling at a particular instant.
07/04/2019

5. Velocity & Acceleration
Velocity is defined as the rate of which displacement changes over time.
The higher the velocity, the faster a body is moving. It is a vector quantity i.e. it requires both magnitude and direction.
Velocity can be zero also if the total displacement is zero, and this is only when the body after travelling a certain distance in any direction comes to rest at the same point where it started.
Velocity is the derivative of displacement with respect to time.
ʋ = 𝑑𝑥 𝑑𝑡 𝑎 = 𝑑ʋ 𝑑𝑡 = 𝑑 2 𝑥 𝑑𝑡 2
Acceleration is the rate of change of velocity with time. A body with a positive acceleration is gaining velocity over time. A body with a negative acceleration is losing velocity over time.
07/04/2019

6. Operations on Vectors We can also subtract one vector from another:
First we reverse the direction of the vector we want to subtract,
Then add them as usual
The most common way is to first break up vectors into x and y parts
The vector a is broken up into the two vectors ax and ay
The magnitude of a vector is shown by two vertical bars on either side as |a|
We use Pythagoras' theorem to calculate it: |a| = √( x2 + y2 )
A vector with magnitude 1 is called a Unit Vector.
07/04/2019

7. Dot Product
Two types of vector multiplications are defined, dot product and cross product.
The scalar product or dot product A.B of two vectors A and B is not a vector, but a scalar quantity (a number with units) equal to the product of the magnitudes of the two vectors and the cosine of the smallest angle between them.
In terms of the Cartesian components of the vectors A and B the scalar product is written as:
A.B = AxBx + AyBy + AzBz » A.B = ABcosϕ
The scalar product is commutative, i.e. A.B = B.A.
Examples: Work, Power
When we form the scalar product of two vectors, we multiply the parallel component of the two vectors.
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8. Cross Product
The vector product or "cross product" of two vectors A and B is a vector C, defined as C=A×B equal to the product of the magnitudes of the two vectors and the sine of the angle between them. (A×B = Absinϕ .ň)
We can find the Cartesian components of C=A×B in terms of components of A and B.
Cx=AyBz-AzBy , Cy=AzBx-AxBz , Cz=AxBy-AyBx
Consider two arbitrary vectors A and B.
Then A = (Ax, 0, 0) and B = (Bx, By, 0) and so, C=Cz=AxBy
The vector product is not commutative, i.e. A×B ǂ B×A
Examples: Torque, Angular momentum
When we form the vector product of two vectors, we multiply the perpendicular component of the two vectors.
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9. Direction
The direction of C can be found by inspecting its components or by using the right- hand rule.
Let the fingers of your right hand point in the direction of A. Orient the palm of your hand so that, as you curl your fingers, you can sweep them over to point in the direction of B. Your thumb points in the direction of C.
If A and B are parallel or anti-parallel to each other,
then C= A×B = ABsinϕ = 0, since sin0° or 180°=0.
If A and B are perpendicular to each other,
then C= A×B = ABsinϕ = 1 since sin90°=1
C has its maximum possible magnitude at 90°.
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10. Torque
Torque, or moment of force is rotational analogue of force. Just as a linear force is push or pull, a torque can be a twist to an object & determines angular acceleration.
Definition: ‘Turning effect of force about a fixed point, as a result of applied force’.
SI Unit: N.m
The point of rotation is called fulcrum or pivot.
The magnitude of torque depends on three quantities: the applied force, moment arm (a vector from origin of coordinate system to the point where force is applied), and angle between force & moment arm. In symbols:
τ = r × F and τ = | r || F |sin θ
Where, r is the position vector and F is the force.
If, F || r, τ is zero, as sin of 0 or 180 is zero.
If, F | r, τ is maximum, as sin of 90 is 1.
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11. Direction
The torque may be clockwise or anticlockwise. It can be determined using right hand rule. Fingers point to direction of moment arm, And are curled to the direction of the force. If turning is anticlockwise, torque is directed upwards & is +ive. If turning is clockwise, torque is downwards & is -ive.
07/04/2019

12. Equilibrium
Equilibrium is the condition of a system in which all the acting forces are balanced. In this state the net effect of all the forces is nil, so that the acceleration is zero.
The state of an object when under the action of forces acting together, there is no change in translational and rotational motion, is called equilibrium.
Static Equilibrium: Body at rest.
Dynamic Equilibrium: Body in uniform motion.
Translational Equilibrium:
At rest or motion in straight line with constant speed.
Rotational Equilibrium:
Rotation about a fixed axis with constant angular speed.
Concurrent forces: Whose line of action passes through same point.
07/04/2019

13. Conditions of equilibrium
Net Force Must Be Zero.
The vector sum of all the forces acting on the object must be zero. Therefore all forces balance in each direction. (Translational Equilibrium, no linear acceleration)
Mathematically, this is stated as ∑Fnet = ma = 0.
∑Fx = 0 , ∑Fy = 0
Net Torque Must Be Zero.
The vector sum of all the torques acting on the object must be zero. (Rotational Equilibrium, no rotational acceleration)
Mathematically, ∑τ = 0
All the moments must balance each other i.e. ∑τ clockwise = ∑τ anticlockwise
The object must be in state of rest or constant acceleration to be in state of equilibrium.
07/04/2019