2020 WRITEN BY ISLAM SHAKER STEM-OBOUR Notes Static Friction: It is the friction experienced by a body when it is at rest. Or in other words, it is the friction when the body tends to move. Dynamic Friction: It is the friction experienced by a body when it is in motion. It is also called kinetic friction. Limiting Friction: It is a maximum value of frictional force, which comes into play, when a body just begins to slide over the surface of the other body. Normal Reaction: It is a force that acts perpendicular to the plane. Angle of Friction: It is an angle of inclined plane, at which a body just begins to slide down the plane. This is also equal to the angle, which the normal reaction makes with the vertical. Coefficient of Friction: It is the ratio of limiting friction to the normal reaction, between the two bodies, and is generally denoted by μ. Laws of Static Friction: - The force of friction always acts in a direction, opposite to that in which the body tends to move. - The force of friction depends upon the roughness of the surfaces. Laws of Dynamic Friction: - The force of friction always acts in a direction, opposite to that in which the body is moving. - For moderate speeds, the force of friction remains constant. But it decreases slightly with the increase of speed. 1|Page All Laws Coefficient of Friction: π= π πΉ = πππ ∅ ∅ = Angle of friction. π = Limiting friction. πΉ = Normal reaction between the two bodies. Equilibrium of A Body on A Rough Horizontal Plane: π = ππ π = Coefficient of friction. πΉ = Normal reaction. π = π. cos π πΉ = π€. π sin π If the force that acts on the body make angle with it π· = The force that acting on the body. Equilibrium of A Body on A Rough Inclined Plane Subjected to A Force Acting Along the Inclined Plane Minimum force (P1), when the body is at the point of sliding downwards. In this case, the force of friction (πΉ1 = π. π 1 ) will act upwards. π·π = πΎ πππ πΆ – π. πΉπ π·π = πΎ × 2|Page π¬π’π§(πΆ−∅) ππ¨π¬ ∅ Maximum force (P2), when the body is at the point of sliding upwards. In this case, the force of friction (πΉ2 = π. π 2 ) will act downwards. π·π = πΎπ¬π’ π§ πΆ + π. πΉπ π·π = πΎ × π¬π’π§(πΆ+∅) ππ¨π¬ ∅ Equilibrium of A Body on A Rough Inclined Plane Subjected to A Force Acting Horizontally Minimum force (P1), when it is at the point of sliding downwards. In this case, the force of friction (πΉ1 = π. π 1 ) will act upwards. π·π πππ πΆ = πΎ πππ πΆ – π πΉπ π·π = πΎ πππ (πΆ – ∅) Maximum force (P2), when it is at the point of sliding upwards. In this case, the force of friction (F2 = μ. R 2 ) will act downwards. π·π πππ πΆ = πΎ πππ πΆ + π πΉπ π·π = πΎ πππ (πΆ + ∅) 3|Page Equilibrium of A Body on A Rough Inclined Plane Subjected to A Force Acting at Some Angle with The Inclined Plane Minimum force (P1), when it is at the point of sliding downwards. In this case, the force of friction (πΉ1 = π π 1 ) will act upwards. π·π πππ π½ = πΎ πππ πΆ – π πΉπ π·π = πΎ × π¬π’π§(πΆ−∅) ππ¨π¬(π½+∅) Maximum force (P2), when it is at the point of sliding upwards. In this case, the force of friction (πΉ2 = π π 2 ) will act downwards. π·π πππ π½ = πΎ πππ πΆ + π πΉπ π·π = πΎ × 4|Page π¬π’π§(πΆ+∅) ππ¨π¬(π½−∅) The Distance Between Two Points in Space: π¨π© = √(ππ − ππ )π + (ππ − ππ )π + (ππ − ππ )π The Norm of The Vector: It is the length of the directed line segment that represents the vector. ββ β = √(π¨π )π + (π¨π )π + (π¨π )π βπ¨ Equality of Two Vectors in Space: β then, π¨π = π©π π¨π = π©π π¨π = π©π If π΄ = π΅ The Unit Vector: It is a vector whose norm equal the unit length. βββββπ¨ = πΌ ββ π¨ ββ β βπ¨ Scalar Product of Two Vectors (Dot Product): βπ¨ β . βπ© β = βπ¨ ββ ββπ© ββ βππππ½ = π¨π π©π + π¨π π©π β .B β = B β .A β A 2 βA . βA = βA ββ iΜ . iΜ = 1 jΜ. jΜ = 1 kΜ . kΜ = 1 The Angle Between Two Vectors: ππππ½ = 0° ≤ θ ≥ 180° 5|Page βπ¨ β .π© ββ ββ ββπ© ββ β βπ¨ The Component of A Vector on The Direction of Another Vector: ββ βππππ½ = π¨π© = βπ¨ βπ¨ β .π© ββ ββ β βπ© Vector Product of Two Vectors (Cross Product): βπ¨ β × βπ© β = (βπ¨ ββ ββπ© ββ β ππππ½)πͺ β πΜ βπ¨ β × βπ© β = | π¨π π©π πΜ π¨π π©π Μ π π¨π | π©π (π π± π π² − π π² ππ± + π π² ππ³ − ππ³ π π² + π π± ππ³ − ππ³ ππ± ) βA × βB = − βB × βA βA × βA = βB × βB = π β Μ , πΜ × π Μ = πΜ , π Μ × πΜ = πΜ πΜ × πΜ = π Μ = −πΜ , π Μ × πΜ = −πΜ , πΜ × πΜ = −π Μ / πΜ × π β × βπ© β = (βπ¨ ββ ββπ© ββ β ππππ½) = βπ© ββ β × π³ When, βπ¨ Then, πΏ = βπ΄β π πππ Cross Product = the area of the parallelogram = double the area of triangle The scalar triple product: π¨π βπ¨ β . βπ© β × βπͺ = |π©π πͺπ π¨π π©π πͺπ π¨π π©π | πͺπ ββ . (π© ββ × πͺ β)= π© ββ . (πͺ β ×π¨ ββ ) = πͺ β . (π¨ ββ × π© ββ ) π¨ Triple Product = the volume of Parallelepiped 6|Page Work: If a force P, acting on a body, causes it to move through a distance S Then, work done by the force P πΎ = π· πππ π½ × πΊ π π±ππππ = π π΅ . π π Graphical Representation of Work: Work done by the Frictional Force: π = −ππ . πΊ If the force and distance are parallel, the amount of work is positive, but if the two vectors are anti-parallel, then the work is negative. Work done by a Gravitational Force: π = −π . π . πΊ Hooke’s Low: π = −π . βπ 7|Page Springs in Parallel: ππ + ππ = Springs in series: π ππ = βπ π βππ / ππ = π βππ Work done by Kinetic Energy: π= π π π π ππ π − π ππ π π Kinetic energy: π€= π π¦ π―π π Power: π·= π π = π. π π = π .π 1 watt = 1 joule / s Efficiency: it is the ratio of work output to work input π= πΎππππππ πΎπππππ × πππ % Energy: it is the capacity to do work. Its unit is (Joule) Work Net Done on The Object: π π π π ππππ = πππ − πππ π = βππ¬ KINETIC ENERGY: It is the energy, possessed by a body, for doing work by virtue of its mass and velocity of motion. π π²π¬ = πππ ( π± = ππ. ππ /ππ ) π 8|Page Work Done on The Object by The Gravity: ππ = −πππ Work Done by The Nonconservative Forces: πππ = βππ¬ + πππ Gravitational Potential Energy: It is the energy possessed by a body, for doing work, by virtue of its position. π·π¬ = π . π = πππ ( π±) Conservation of Mechanical Energy: When a physical quantity is conserved the numeric value of the quantity remains the same throughout the physical process; its final value is the same as its initial value. π¬ = π²π¬ + π·π¬ The Force Exerted by The Spring: ππ = −π π The Average Force: π= −π π π The force ππ is often called a restoring force because the spring always exerts a force in a direction opposite the displacement of its end, tending to restore whatever is attached to the spring to its original position. Work Done by The Spring: π π π π ππ = − ( π π π π − π π π π ) Spring Potential Energy: π π·π¬π = π ππ π 9|Page 10 | P a g e
