Motion And Forces

Physics 100 Unit 1:

Motion and Forces

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Mathematical Concepts Physics:

a branch of science concerned with the ultimate laws which govern the structure of the universe and forms of matter and energy and their interactions. Physics has developed out of the efforts of men and women to explain our physical environment. These efforts have been so successful that the laws of physics now encompass a remarkable variety of phenomena, including planetary orbits, radio and TV waves, Magnetism and lasers, to name a few.

Units of Measurement SI Units

stands for the French phrase “ Le Systeme International d’ Unites.” This system employs the meter (m) as the unit of length, the kilogram (Kg) as the unit of mass, and the second (s) as the unit of time.

CGS System

uses the centimeter (cm), gram (g) and the second (s), for length, mass and time, respectively. 2

BE System

British Engineering system (the gravitational version) uses the foot (ft), the slug (sl) and the second (s).

Units of Measurement

meter

the distance that light travels in a vacuum in a time of 1/299 792 458 seconds.

kilogram

the mass of a standard cylinder of platinum-iridium alloy kept at the International Bureau of Weights and Measures.

second

the time for a certain type of electromagnetic wave emitted by cesium-133 to undergo 9 192 631 770 wave cycles.

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Some Conversion Factors Length:

Force:

1 1 1 1 1 1

1 lb = 4.448 N 1 N = 105 dynes = 02248 lb

in = 2.54 cm ft = 0.3048 m mi = 5,280 ft = 1.609 km m = 3.281 ft km = 0.6214 mi angstrom (Å) = 10-10 m

Mass:

Work and Energy: 1 1 1 1

J = 0.7376 ft-lb = 107 ergs Btu = 1055 J kWh = 3.60 x 10-6 J eV = 1.602 x 10-19 J

1 slug = 14.59 kg 1 kg = 100 grams = 6.852 x 10-2 slug 1 atomic mass unit (u) = 1.6605 x 10-27 kg(1 kg has a weight of 2.205 lbs where acceleration due to gravity is 32,174 ft/s2) ) Time: Power: 1 day = 24 hrs = 1.44 x 103 min = 8.64 x 104 s 1 yr = 365.24 days = 3.156 x 107 s

1 hp = 550 ft-lbs = 745.7 W 1 W = 0.7376 ft-lbs

Speed:

Angle

1 mi/hr = 1.609 km/ hr 1 radian = 57.300 1 km/hr = 0.6214 mi/hr = 0.2778 m/s I0 = 0.01745 radian 4

Conversion of Units

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Solve the following: + 1. How many seconds are there in (a) one hour and thirty five minutes, and (b) one day? 2. The largest diamond ever found had a size of 3106 carats. One carat is equivalent to a mass of 0.200 g. Given that one kg is equal to 2.205 lb, find the weight of the diamond in pounds. 3.How many square meters are there in 1330 ft2 ?

Trigonometry

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Ex. 3 A building casts a shadow 67.2 m long. The angle between the sun’s rays and the ground is θ = 500. Find the height of the building. θ = 50o ; ha = 67.2 m

Given:

ho

h θ = 50 ha = 67.2

tan θ = 0

ho / ha

ho = tan θ x ha = tan 500 x 67.2 = 1.192 x 67.2 = 80.1 m Ans.

Inverse Trigonometric Functions:

θ = sin-1 (ho/h) θ = cos-1 (ha/h) θ = tan-1 (ho/ha)

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Ex. 4 A lake front drops off gradually at an angle θ. A lifeguard rows straight out from the shore 14.0 m and drops a weighted fishing line. By measuring the length of the line, the lifeguard determines the depth to be 2.25 m. (a) (b)

what is the value of θ ? what would be the depth 22 m from the shore?

ha = 14 m

(a):

θ

ho = 2.25 m

θ = tan-1 (ho/ha)

(b):

= tan-1 (2.25/14) = tan-1 (0.1607) = 9.13O Ans. h’o = 2.25 x (22/14) = 3.54 m Ans.

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Scalars and Vectors

Displacement

a change of position of a particle

Scalar Quantity

one that can be described with a single number (including any units) giving its size and magnitude, e.g., temperature, mass, time, length, density and time.

Vector Quantity

a quantity that deals inherently with both magnitude and direction. Some quantities that are vectors are force, velocity, acceleration, momentum, the electric field, and the magnetic field.+

Negative of a Vector

another vector of equal magnitude but opposite direction

The fact that a quantity is positive or negative does not necessarily mean that the quantity is a scalar or a vector.

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Vector Addition and Subtraction ADDITION When two vectors are colinear, i.e., they point along the same direction, the total vector or resultant vector R is given by: R = A + B A

B R

When two vectors are perpendicular, the resultant vector R is found by:

θ

R

B

R = ( A2 + B2 )1/2

A

θ = tan-1 (B/A) 11

When the vectors are neither colinear nor perpendicular, the resultant vector R is found by graphical technique. A diagram is constructed in which the arrows are drawn tail to head. The lengths of the vector arrows are drawn to scale and the angles are drawn accurately. Then, the length of the arrow representing the resultant vector is measured with a ruler. The length is converted to the magnitude of the resultant vector by using the scale factor with which the drawing is constructed.

R

B A

R is measured with a ruler and converted to the resultant vector by using the scale factor. SUBTRACTION When a vector is multiplied by -1, the magnitude of the vector remains the same, but the direction of the vector is reversed. Vector subtraction is carried out exactly like vector addition, except that one of the vectors added is multiplied by -1. 12

The Components of a Vector VECTOR COMPONENTS In two dimensions, the vector components of a vector A are two perpendicular vectors Ax and Ay, that are parallel to the x and y axes, respectively, and add together vectorially so that A = Ax + Ay. +y

A Ay

A

Ax

+x

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+y

Addition of Vectors by Components C A

B

By Bx Ay

Ax

C=A+B C = Cx + Cy

+x

Cx = Ax + Bx Cy = Ay + By Example: A man runs 145 m in a direction 200 east of north (displacement vector A) and then 105 m in a direction 350 south of east (displacement vector B). Determine the magnitude and direction of the resultant vector C for these two displacements.

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Diagram: Ax

Bx 350

B = 105m

Ay

A=145m

By

C 200

C=A+B C = Cx + Cy

C = √(Cx2 + Cy2) Cx = Ax + Bx = 49.6 + 86.0 = 135.6 m Cy = Ay + By = 136.3 + (-60.2) = 76.1 m

Ax = Asin200 = (145m) 0.3420 = 49.6 m Bx = Bcos350 = (105m) 0.8191 = 86.0 m

Ay = Acos200 =(145m) 0.9397= 136.3m By = Bsin 350 = - (105) 0.5736= - 60.2m

C = √ (135.6)2 + (76.1)2 = 155.5 m Answer

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Solve: The figure below shows two displacement vectors A and B, which add together to give a resultant displacement C. Find the magnitude and direction of C. +y

C

B (62 m) 35

0

+x

A (47 m)

Does the Pythagorean theorem apply directly to the problem? Does the component method for vector addition apply to this problem, even though the vectors are not perpendicular? +

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Solve. •A highway is to be built between two towns, one of which lies 30 km south and 72 km west of the other. What is the shortest length of highway that can be built between the two towns, and at what angle would this highway be with respect to due west? •Displacement vector A points due east and has a magnitude of 2.00 km. Displacement vector B points due north and has a magnitude of 3.75 km. Displacement vector C points due west and has a magnitude of 2.50 km. Displacement vector D points due south and has a magnitude of 3.00 km. Find the magnitude and direction (relative to due west) of the resultant vector A + B + C + D. •On take-off, an airplane climbs with a speed of 180 m/s at an angle of 340 above the horizontal. The sun is shining directly overhead. How fast is the shadow of the plane moving along the ground? • On a safari, a team of naturalists set out toward a research station 4.8 km away in a direction 420 north of east. After travelling in a straight line 2.4 km, they stop and discover that they have been travelling 220 north of east, because their guide misread his compass. What are (a) the magnitude and (b) direction relative to due east of the displacement vector now required to bring the team to the research station?

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