Laws of Motion

Laws of motion are laws that establish relationship between force and motion. Force is a physical quantity that changes the motion or the shape of an object. Or in layman terms, force is a push or a pull. The SI unit of force is the Newton abbreviated as N. There are three laws that establish relationship between force and motion which were discovered by the British scientist Isaac Newton.

Newton's First Law states that an object stays at rest or continues to move in a straight line with a constant speed unless acted upon by a net force. This law describes what happens to an object when there is no force acting on it. A net force is required to change the direction of motion or to accelerate an object. No force is required to keep an object going in a straight line with a constant speed. There must be a net force acting on an object moving in a circular path with a constant speed because its direction is changing. There must be a net force acting on an object falling in a straight line because its speed is changing. It is a common experience that when a driver applies the brakes suddenly, his (her) body tends to move forward. This is because the force exerted by his (her) back sit ceases to act after the brakes are applied and his (her) body tends to move in a straight line with the same speed he (she) had when the brakes were just applied. Of course other forces will stop this motion shortly.

Newton's second Law states that the net force acting on an object is directly proportional to its acceleration. The constant of proportionality between the force acting on an object and its acceleration is a measure of the amount of matter the object has and is called the mass of the object.

Fnet = ma

where Fnet is the net force acting on an object of mass m moving with acceleration a. The SI unit of mass is the kilo gram (kg). The unit of force (N) is equal to the product of the unit mass (kg) and the unit of acceleration (m/s2).

Example : Calculate the acceleration of an object of mass 20 kg when acted upon by a force of 100 N.

Solution: Fnet = 100 N; m = 20 kg; a = ?

Fnet = ma

a = Fnet /m = (100/20) m/s² = 5 m/s²

Example : A force that causes an acceleration of 3 m/s² when acting on object A of mass 5 kg is acting on object B of mass 30 kg. Calculate the acceleration of object B.

Solution: mA = 5 kg; aA = 3 m/s²; mB = 30 kg; Fnet = ?

Fnet = mAaA = 5*3 N = 15 N

Fnet = mBaB

aB = Fnet /mB = 15/30 m/s² = 0.5 m/s²

Newton's third law states that for any action there is an equal but opposite reaction. If object A exerts a force on object B, then object B exerts a force on object A which is of the same magnitude but opposite in direction.

FAB = -FBA

where FBA is force exerted on object B by object A and FAB is force exerted by on object A by object B. Action reaction forces act on different objects. An object can not exert force on itself directly. An object can exert force on itself only by exerting force on another object so that the reaction force acts on it. If one wants to walk forward, he (she) has to push on the ground backward so that the reaction force of the ground pushes him (her) forward. If one wants to swim forward, he (she) has to push on the water backwards so that the reaction force of the water pushes him (her) forward. If a car is to move forward, the wheels of the car must push on the ground backwards so that the reaction force of the ground pushes the car forward. If a rocket is to be propelled into space, it has to push on a gas downward so that the reaction force of the gas propels it upward.

Example : Object A exerts a force of 8 N east on object B. Determine the magnitude and the direction of the force exerted by object B on object A.

Solution: FBA = 8 N east; FAB = ?

FAB = -FBA = 8 N west

Types of Forces

Forces may be generally classified as contact and non-contact forces.

Non-contact Forces

Non-contact forces are forces that objects exert on each other without a direct contact between the objects. For example earth can attract an object in its vicinity towards itself even though there is no direct contact between earth and the object. This kind of force is called gravitational force. Other non-contact forces include electrical and magnetic forces. In this course, we will deal only with gravitational force.

Weight

Weight of an object in the vicinity of a massive object is the gravitational force exerted by the massive object on the object. For example, the weight of an object in the vicinity of earth is the gravitational force exerted on the object by earth. The acceleration of a falling object, which is the acceleration due to the gravitational force exerted by earth, is a constant and is equal to |g| = 9.8 m/s². Therefore the weight of an object on the vicinity of earth is equal to the product of its mass and this gravitational acceleration.

w = m|g|

where w is the weight of an object of mass m. In the vicinity of earth, weight and mass are proportional; which makes it possible to measure mass by measuring weight. But mass and weight are very different physical properties. Mass is a measure of the amount of matter the object has. It is an inherent property of the object. Its value does not depend on other objects in its vicinity. An object will have the same mass on all planets. On the other hand, weight is a measure of the gravitational force due to massive objects in its vicinity. Its value depends on the mass of the massive object in its vicinity. An object will have different weights on different planets. For example the weight of an object on the moon is one sixth of its weight on earth, because gravitational acceleration on the moon is one sixth of that on earth. The unit of weight is the unit of force which is the Newton.

Example : Calculate the weight of a 100 kg object.

Solution: m = 100 kg; w = ?

w = m|g| = 100*9.8 N = 980 N

Example : An object of weight 100 N is being acted upon by a net force of 20. Calculate its acceleration.

Solution: w = 100 N; Fnet = 20 N; a = ?

w = m|g|

Contact Forces

Contact forces are forces exerted between objects in direct contact with each other. These kind of forces can be identified easily by identifying the objects in contact with the object under consideration. But there is one kind of contact force that deserves a special attention. This is force between two surfaces in contact.

Surface Forces

Surface forces are forces between surfaces in contact. A surface force is customarily decomposed into a component that is parallel to the surfaces and a component that is perpendicular to the surfaces. The component that is parallel to the surfaces is called the force of friction. When the two surfaces slide on each other, this force opposes the motion. The direction of this force on one of the surfaces is opposite to the direction of the sliding motion with respect to the other surface. The component that is perpendicular to both surfaces is the force that presses the two surfaces together and is called the normal force. Without this force, the two surfaces will not be in contact. It is a common experience that the greater the force that presses the two surfaces together (the normal force), the more difficult it is to slide one of the surfaces relative to the other surface. In other words, the greater the normal force, the greater the force of friction. Experiment shows that the force of friction and the normal force are directly proportional. The constant of proportionality is called the coefficient of friction between the two surfaces. It is unit less.

f = μN

where f is the force of friction, N is the normal force and μ is the coefficient of friction between the surfaces. There are two kinds of coefficients of friction. Experiment shows that the amount of force required to just get an object sliding is greater than the force required to keep it sliding once it has started sliding. This means two types of coefficient of friction are required. The coefficient of friction related with the force of friction just before it starts sliding is called the static coefficient of friction.

f = μsN

where μs is the static coefficient of friction between the surfaces. The coefficient of friction related with the force of friction when the two surfaces are sliding with respect to each other is called the coefficient of kinetic friction.

f = μkN

where μk is the kinetic coefficient of friction between the surfaces. Since the force of friction just before an object starts sliding is greater than the force after it starts sliding, coefficient of static friction is greater than coefficient of kinetic friction.

μs > μk

Coefficient of friction depends on the types of surfaces in contact. For example sliding on a rough surface is much harder than sliding on a smooth surface. Coefficient of friction does not depend on the area of the surface area of contact. A rectangular solid object may have surfaces of different surface areas. All of the surfaces will have the same coefficient of friction. Also, coefficient of friction does not depend on the relative speed between the surfaces. Whether the relative speed between the surfaces is 10 m/s or 20 m/s coefficient of friction will always be the same.

Example : An object of mass 10 kg is sliding in a horizontal surface with a uniform speed. The coefficient of kinetic friction is 0.2.

  1. Calculate the normal force pressing the two surfaces together.

    Solution: m = 10 kg; μk = 0.2; N = ?

    The net force acting on the object in the direction perpendicular to the surfaces must be zero, because the object is not moving in this direction. There are two forces acting on this object in this direction. One is the weight of the object which is directed downward. The other force is the vertical component of the force exerted by the surface on which it is sliding. This is the normal force. Since the net force in this direction is zero, these two forces must be balancing each other.

    N = w = m|g| = 10*9.8 N = 98 N

  2. Calculate the force of friction.

    Solution: f = ?

    f = μkN = 0.2*98 N = 1.96 N

Example : An object of mass 10 kg is being pulled on a horizontal surface with a horizontal force of 100 N. The coefficient of kinetic friction is 0.3.

  1. Calculate the normal force pressing the two surfaces together.

    Solution: m = 10 kg; F = 100 N (F is the horizontal force pulling to the right); μk = 0.3; N = ?

    The net vertical force acting on the object must be zero because the object is not moving in this direction. Therefore the vertical forces acting on this object (normal force and its weight) must balance each other.

    N = w = m|g| = 10*9.8 N = 98

  2. Calculate the force of friction resisting the motion of the object.

    Solution: f = ?

    f = μk N = 0.3*98 = 29.4 N

  3. Calculate the acceleration of the object.

    Solution: a = ?

    Horizontally, there are two forces acting on the object. One is the horizontal force pulling on the object (F). The other is the force of friction (f) opposing this force. The net horizontal force is the difference between these forces.

    Fnet = F - f = (100 - 29.4) N = 70.6 N

    Fnet = ma

    a = Fnet /m = 70.6/10 m/s² = 7.06 m/s²

Example: Consider a book on the top of a table. Identify four action reaction pairs.

Solution:

  1. Gravitational force exerted by earth on book and gravitational force exerted by book on earth.

  2. Gravitational force exerted by earth on table and gravitational force exerted by table on earth.

  3. Surface force exerted by table on book and surface force exerted by book on table.

  4. Surface force exerted by ground on table legs and surface force by table legs on ground