Lecture 7 Forces and the motion they produce As we have seen in the last 6 lectures we use displacement velocity and acceleration to describe the motion of an object The study of this motion is called kinematics Now we begin a study of the forces and the motion that they produce The methods we have used in studies of kinematics are essential to our work but now we add another feature the idea that forces can produce motion To get started we need to talk about forces Forces What are they Forces are traditionally subdivided into field forces and contact forces Examples of field forces are gravity and the coulomb force between charged particles Examples of contact forces are friction or a hammer hitting a nail However all forces are really field forces if you look closely enough In fact there are four fundamental field forces the gravitational force the electromagnetic force the weak force the strong force The strong force is what holds the nucleus together and the weak force also acts at the nuclear length scale about a fermi The forces that dominate our everyday life are the gravitational and electromagnetic force and the forces that are called contact forces are really due to the quantum mechanics of the electromagnetic force of electrons in atoms To understand dynamics it is very useful to develop the methods based on the idea of contact forces so we don t have to worry about the details of the motion of electrons near atoms Since force is so important we give it a separate unit the Newton N This is a derived unit as it is related to the base units of mechanics m kg s We will see the relation below The effect of forces Newton s laws Note that there is no way to derive these laws They are laws which were postulated in order to explain experimental observations They have stood the test of time and and now considered fundamental to all physics They were written down most completely in Newton s Principia actually it s full title is Philosophiae Naturalis Principia Mathematica published in 1687 in latin Principia remains the most profound scientific treatise ever published with Darwin s Origin of Species published in 1859 an honorable second First law 1 A body in motion with velocity v remains in motion with velocity v unless acted upon by an net external force Second Law P A net external force F acting on an inertial mass m produces an P acceleration a F m This is more commonly written as X F m a 1 From this equation we see that the unit of force N kgm s2 Note that 1N 0 225lb where the lb is the unit of force in US units The unit of mass in US units is the slug Third Law To every action there is an equal and opposite reaction Applications The number of applications of the above simple looking laws is truly staggering and is testament to the fact that there are simple unifying principles underlying much of the behavior we observe in world around us However though the above laws look simple solving problems using them can be complex and requires a good physical understanding of what Newton s laws tell us In fact there are many applications of even the simple case where there is P no motion ie a 0 In that case Newton s second law tells us that F 0 so there can be no net force on the object This case is extremely important in civil engineering where we want bridges and buildings to stay where they are The study of cases where the forces sum to zero is called statics or engineering statics This case also has many biomechanics applications for example in hip replacements and artificial knees Dynamics is the study of the case where a 6 0 and is extremely important in mechanical engineering where the design of machines to move and do work is a key task As we shall note later Newton s laws also underlie fluid dynamics such as blood flow in the body and atmospheric dynamics such as weather patterns and hurricanes First lets look at an application to one dimensional motion Dynamics example Consider a stationary mass m 10kg on a frictionless surface located at the origin At t 0 we apply a constant force F 2 Fx Fy Fz 6 0N 0 0 to the mass Find the position of the mass 5 0s after the force is applied Solution From Newton s second law we have a P 6N 0 0 F m 10kg 2 Since the acceleration is only in the x direction the mass moves only in the x direction so that ax 6 10m s2 Since the force is constant so is the acceleration so we can use our constant acceleration formula 1 x v0x t ax t2 0 50 0 60 5 02 m 0 75m 2 3 The mass is located at x 0 75m at time t 5s Now lets look at an application in statics Statics example Consider a gymnast who holds herself in the air while keeping her arms vertical by using her two hands to grasp two different rings The rings are at the same height and are attached to a supporting beam by two different ropes The first rope makes an angle of 200 to the vertical and the other rope makes an angle of 400 to the vertical and the two ropes lie in the same plane Find the tension in the two ropes if the gymnast has a mass of 40kg Solution Since there is no motion the sum of the forces must be zero In the x direction this implies that T1 sin 200 T2 sin 400 4 The sum of the forces in the y direction gives T1 cos 200 T2 cos 400 40kg 9 8m s2 5 From the first equation we have T2 T1 sin 200 sin 400 0 53T1 Putting this in the second equation we get 1 35T1 392N 6 so that T1 291N Using this in either of the above equations yields T2 155N 3
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