A micro-course in dynamics that is section two of the course in classical mechanics available on Grade-12 Physics: Mechanics and Physics 12: Classical Mechanics . In this course there is an introduction to the Newton's laws of motion, and lectures in different topics of dynamics which are discussed in grade-12 physics. Problems and solutions are also included to this course.

Newton’s Laws of Motion



An object at rest or in

- Second law: When an external force acts on the object, the object moves with an acceleration that is proportional to the force and is in the same direction as the direction of the force. The equation between the sum of external forces acting on the object (or the net force acting on the object) and its acceleration can be written as, - First law, the Law of Inertia (resistance to the change in motion):An object at rest or in uniform motion remains at rest or uniform motion unless an external force acts on it.- Second law: When an external force acts on the object, the object moves with an acceleration that is proportional to the force and is in the same direction as the direction of the force. The equation between the sum of external forces acting on the object (or the net force acting on the object) and its acceleration can be written as,





in this equation, 'm' mass of the object, is in fact the proportionality constant.

- Third law:

For every action force there is a reaction force that is equal in size, but opposite in direction.

Friction

Frictional forces appear between two surfaces which are in contact. Frictional force is static, if the two surfaces are at rest relative to each other, and it is kinetic if the two surfaces are moving relative to each other. For more complete discussion watch the video below:

Tension

When two objects are connected to each other by a string, they exert forces on the string, and string itself exerts forces on the objects. The magnitude of the force exerted on (and by ) a string is called tension. In order to find the amount of tension that a string, rope, or cable must be able to withstand, you need to apply Newton's laws of motion.

Physics Mechanics Course for High School Students, 65% off with coupon code XNA4NNRY9M, expires on 20/12/2019:

Motion along an Incline

When an object is moving on an inclined surface, there are three forces acting on it: gravitational force, kinetic frictional force that is opposite to the direction of the motion of the object, and normal force exerted by the inclined surface and it is perpendicular to the inclined surface.

If the angle of incline is θ, the components of gravitational force on inclined surface and perpendicular to it, are "mg sinθ" and "mg cosθ", respectively. Newton's second law on inclined surface, and perpendicular to the inclined surface, can be written as,

where "N" is the normal force, "a" is the acceleration of object, and,

is the frictional force acting on object, where μ

k is the coefficient of kinetic friction. So the equation of motion on inclined surface can be written as,

Atwood Machine

Atwood machine is made of two objects connected by a string that passes over a pulley and the pulley is hung from a ceiling. If you ignore the frictional forces, there are only two forces acting on the objects: gravitational force, and tension ( exerted by the string on the objects).

According to the picture above, equations of motion of these two objects (or Newton's second law), can be written as,

where "T" is the tension exerted by the string on the objects, "

m 1 g"

and

" m 2 g" are gravitational force upon the objects, and "a" is the absolute value of acceleration of the objects, and since

"m 1 " is smaller than "

m 2 ", it moves upward, while

" m 2 " moves downward.

By subtracting second equation from the first one, we get,

and,

By inserting "a" in one of equations (1) or (2), we get,

Hooke's Law

When an object is connected to a spring, the force that it applies on the spring, extends or compresses the length of the spring. Spring itself exerts a reaction force on the object that is called restoring force.

The applied force

is proportional to the value of extension or compression of the spring , and it is given by,

where "x" is the amount of extension or compression of the spring, and

"κ" is the spring constant.

According to the Newton's third law the amount of restoring force is equal to the amount of applied force, but its direction is opposite to the direction of applied force, i.e.,

Free Fall

A falling object is in free fall if gravity is the only force acting on it, that is given by,





where "m" is the mass of the object, and g is the gravitational acceleration. Gravitational acceleration and so gravitational force are both vectors towards the center of the Earth.

Near the surface of the Earth, if you ignore air resistance, a falling object is in free fall; and the motion of the object can be considered as a uniformly accelerated motion , because "g", the magnitude of the acceleration of the object, is approximately constant in short distances. The average amount of "g" around the Earth, and near the Earth's surface is about g = 9.8 m/s

2

So the equations of motion of an object that is in free fall near the Earth can be found as:

v(t) = -g(t- t 0 ) +v 0





v2 - v 0 2 = -2g(y- y 0 )





y(t) = -( ½) g(t- t 0 )2 +v 0 (t- t 0 ) +y 0





where "v", and "y" are vertical velocity and position of the object, and "t" is the time, with initial values of:

v 0 ,

y 0 ,

t 0

Projectile Motion



Projectile motion is the motion of an object thrown near the earth’s surface, if gravity is the only force acting on it. Projectile moves on a vertical plane. If you define a rectangular two-dimensional coordinate system on the plane, with x-axis on horizon and y-axis in vertical direction, and with origin located on projectile launch point, on x-axis projectile motion is uniform (there isn’t any horizontal force) and on y-axis projectile is in free fall.



Figure 1: Projectile thrown under angle with the horizon,

with initial velocity of v 0 . Its range on x-axis is R,

and its maximum height is H.



Initial values of the position and time (launch point and time) are:

x 0 = y 0 = 0 , and

t 0 = 0

So the equations of motion of projectile on x and y axises are,

From these relations you can find trajectory equation, and maximum height and range of the projectile as,

You can also find time of flight of the projectile (time taken for the projectile to hit the ground again) as,









Apparent Weight and Elevator

The weight of an object on Earth is the gravitational force exerted by the Earth on the object, W= mg ; where ' m ' is the mass of the object and ' g ' is the gravitational acceleration. But the weight that you feel is apparent weight that is the normal force exerted on you by the surface that is in contact with you; this force is the reaction of the force you exert upon surface. So the apparent weight is not always equal to the gravitational force (mg) acting on you. For instance when you are in free fall, you don't exert any force on the surface that is in contact with you; so your apparent weight is zero.

When you are inside an elevator, your apparent weight varies while the acceleration of the elevator varies...... find more

Problems and Solutions

Problem 1

An object with mass 'm' is at rest on an inclined surface that makes angle θ with the horizon.

What is the static frictional force between the object and inclined surface?

Assume that you can increase the angle between inclined surface and horizon; at θ = θ i the object begins to slide downward. Find the coefficient of static friction between the object and inclined surface in terms of θ i .

Problem solving lecture in dynamics:1

Problem 2

A 60kg person is standing on a scale inside an elevator. The elevator is raising at a constant speed but then begins to slow down with a constant acceleration of 0.5 m/s2.

What is the sign of the acceleration? What is the reading on the scale while the elevator is accelerating?

Problem solving lecture in dynamics:2

Problem 3

A student pulls an object with mass of m=10 kg towards the top of a slope inclined with angle of θ = 30 ° . The coefficient of kinetic friction μ k =0.2, and the student and object move with a constant speed. Find the tension force that student applies on the object.

Problem solving lecture in dynamics:3

Problem 4

An object with mass of m 1 =2 kg located on a table and is attached to a light string passing over a pulley and holding an object with mass of m 2 =0.5 kg, suspended in the air.

Ignore the mass of string and find the acceleration of these two objects and tension in the string when the suspended mass is released if the surface of the table is frictionless.

Problem solving lecture in dynamics:4

Static and Kinetic Friction