A micro-course in dynamics that is section two of the course in classical mechanics available on 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
- 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,
- 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:
Static and Kinetic Friction
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.
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.
where "N" is the normal force, "a" is the acceleration of object, and,
is the frictional force acting on object, where μk is the coefficient of kinetic friction. So the equation of motion on inclined surface can be written as,
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,
By subtracting second equation from the first one, 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,
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/s2
So the equations of motion of an object that is in free fall near the Earth can be found as:
v(t) = -g(t- t0) +v0
v2 - v02 = -2g(y- y0)
y(t) = -(½) g(t- t0)2 +v0(t- t0) +y0
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 v0. Its range on x-axis is R,
and its maximum height is H.
and its maximum height is H.
Initial values of the position and time (launch point and time) are:
x0 = y0 = 0 , and t0 = 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=θi
the object begins to
slide downward. Find the coefficient of static friction between
the object and inclined surface in terms of θi
.
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
Problem solving lecture in dynamics:3
Problem 4
An
object with mass of m1=2 kg located on a table and is
attached to
a light string passing over a pulley and holding an object with mass
of m2=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
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