ForceAndMotion <!--Induction_and_Inductance -->

General Physics (calculus based) Class Notes

Dr. Rakesh Kapoor, M.Sc., Ph.D.

Former Faculty-University of Alabama at Birmingham, Birmingham, AL 35294


Force and Motion

Objectives

Up to now we have studied “kinematics” or study of the motion of objects using parameters such as the position vector, displacement, velocity and acceleration.

We have not understood the cause of this motion.

In this chapter we will study the cause of motion or part of mechanics known as “dynamics” and following are our objectives.

Understand Newton’s laws of motion.

Study the relation between a force and the acceleration.

Understand the concept of mass.

Understand the concept of an inertial reference frame.

How to draw a free body diagram?

Understand basic structure of a solid object, this will help us in understanding source of two forces: Normal force and Tension force

Learn about gravitational force and weight.

Learn about force of friction.

Learn to solve problems involving these forces.

Limitation of Newtonian Mechanics

Newtonian mechanics is the study of relation between a force and the acceleration produced by it.

Newtonian mechanics has limitations and is not applicable in following situations.

When the speeds of the interacting bodies are very large (1% or more of the speed of light) the Newtonian mechanics has to be replaced with Einstein's special theory of relativity (1905),

Theory of relativity holds at any speed.

If the interacting bodies are on the scale of atomic structure (for example, they might be electrons in an atom), we must replace Newtonian mechanics with quantum mechanics (1926).

Newton’s First Law

A body will keep moving with constant velocity if no force acts on it.

An object moves in a straight line and at constant speed except in the absence of its interaction with other object.

Force is generated by interaction between objects.

This law only gives conceptual connection between interactions and their effect on the motion of objects.

This law does not allow us to make quantitative predictions.

We can not use this law to predict the exact trajectory of a projectile or uniform circular motion.

Let us analyze it.

Law tells us that force changes the velocity of an object.

Acceleration is change in velocity.

Therefore we can say that Force produces acceleration ForceAndMotion_1.gif.

If force is zero or the object is not interacting with any object, ForceAndMotion_2.gif.  

Newton’s Second Law

Experimentally it was found that acceleration ForceAndMotion_3.gif of an object is proportional to the total force acting on that object.

ForceAndMotion_4.gif

Since acceleration is a vector quantity therefore force ForceAndMotion_5.gif should also be a vector quantity.

ForceAndMotion_6.gif

According to Newton's second Law, the proportionality constant is known as the mass of the object.

ForceAndMotion_7.gif

What is mass?

Mass is an intrinsic characteristic of a body that relates a force on the body to the resulting acceleration.

Mass is a scalar quantity.

You can have a physical sensation of mass only when you attempt to accelerate a body, as in the kicking of a baseball or a bowling ball.

What happens if two or more forces act on an object?

When two or more forces act on an object we can find their "net" force ForceAndMotion_8.gif, or total force, by adding individual forces as vectors.

ForceAndMotion_9.gif

This fact is called the "principle of superposition" for forces.

Newton's Second Law can be re written as

ForceAndMotion_10.gif

In unit vector notation it can be written as

ForceAndMotion_11.gif

Where

ForceAndMotion_12.gif

What kinds of forces are included in net Force?

One or more objects can be considered as a "system"

Everything not in the system is part of the "surroundings"

Net force includes only the forces that the "surroundings" (external objects) exerts on the "system" of interest.

Such forces are called external forces.

In contrast, internal forces are forces that one part of a system exerts on another part of the system.

Internal forces are not included in net force.

Checkpoint - 1

Which of the figure’s six arrangements correctly show the vector addition of forces ForceAndMotion_13.gif and ForceAndMotion_14.gif to yield the third vector, which is meant to represent their net force ForceAndMotion_15.gif?

Units of Force

Force produces acceleration. Thus, a force is measured by the acceleration it produces.

If we apply some force on a body of mass 1 kg kept on a frictionless surface, and this force produces 1 m/ForceAndMotion_17.gif acceleration in that body. The magnitude of that force is defined as 1 N (Newton).

ForceAndMotion_18.gif

Checkpoint - 2

The figure here shows two horizontal forces acting on a block on a frictionless floor. If a third horizontal force ForceAndMotion_19.gif also acts on the block, what are the magnitude and direction of ForceAndMotion_20.gif when the block is (a) stationary and (b) moving to the left with a constant speed of 5 m/s?

ForceAndMotion_21.gif

Inertial reference frame

An inertial reference frame is one in which Newton's law of inertia is valid.

Newton's law involves measurement of force or acceleration.

In the study of relative velocity we made the following observation

When we measure acceleration of an object from two reference frames and the frames are moving with constant velocities. The measured values of acceleration in both the frames, turns out to be same.

This means all the reference frames moving with constant velocity are inertial reference frames.

In other words any non-accelerating object is an inertial reference frame.

All accelerating reference frames are non-inertial.  

Is our earth an inertial reference frame?

We know that earth is rotating around its axis and around sun

ForceAndMotion_22.gif ForceAndMotion_23.gif

A rotating object is an accelerating object; therefore earth is a non inertial reference frame.

Suppose we observe a puck sliding with constant velocity from North pole towards South pole on a long ice strip.

Standing on the earth, its path will observe like the red line shown in Fig. b.

Observed deflection is without any force so, this is violation of Newton's Law and proves that Earth is a non inertial reference frame.

For many practical situations we assume earth to be an inertial reference frame.

Suppose the puck is sent sliding along a short strip.

In this situation the deflection will be negligible and earth can be assumed to be an inertial reference frame.

Free Body Diagram

Part of the procedure of solving a mechanics problem using Newton’s laws is drawing a free body diagram.

First we choose the object to be studies and we call it a “system” of objects (It could be one object or more object).

Draw the system as a point.

Now choose the axes and draw all the external forces acting on the system.

Internal forces are not included. Let us look for horizontal external forces in following examples.

ForceAndMotion_24.gif

ForceAndMotion_25.gif

Fig. (a) System = Block A + Block B. External Force is only ForceAndMotion_26.gif.

Fig. (b) System = Block A. There are two external forces ForceAndMotion_27.gif and ForceAndMotion_28.gif.

Fig. (c) System = Block B. There is one external force ForceAndMotion_29.gif.   

Problem - 1

Two horizontal forces act on a 1.8 kg chopping block that can slide over a frictionless kitchen counter, which lies in a xy plane. One force is ForceAndMotion_30.gif. Find the acceleration of the chopping block in unit - vector notation for each of the following second forces. (a) ForceAndMotion_31.gif (b) ForceAndMotion_32.gif (c) ForceAndMotion_33.gif

The Gravitational Force

Any two objects attract each other with force called gravitational force.

Here when we speak of the gravitational force ForceAndMotion_34.gif on a body, we usually mean a gravitational force that pulls on it directly toward the center of Earth.

Here we assume that the ground is an inertial frame.

If we neglect the effect of air, the only force acting on the body is the gravitational force ForceAndMotion_35.gif.

Suppose a body of mass m is in free fall with the free fall acceleration g.

According to Newton’s second law :

ForceAndMotion_36.gif

Can you answer?

You are given a copper block and you know that it is made up of several copper atoms.

ForceAndMotion_37.gif

Why not these atoms spontaneously evaporate or fall apart?

Why is it very difficult to compress the block and reduce its size?

Atoms and ball - spring model

All matter consists of atoms, whose typical radius is about ForceAndMotion_38.gif meter.

Atoms in all the materials attract each other when they are close to each other but not too close, that is why the metal does not spontaneously evaporate or fall apart from the block.

Atoms repel each other when they get too close to each other, that is why it is very difficult to compress the block, so the atom must resist an attempt to push them closer

Evidently atoms in the block are just at right distance from each other, neither too close nor too far away.

This just right distance is called "equilibrium" distance between atoms.

Atoms and ball - spring model

Consider two balls connected with a spring.

If we try to take these balls apart (Click hold and drag any one ball), the spring will try to bring them close. So we can say balls attract each other when we try to take them apart.

If we try to bring balls closer (click hold and drag any one ball), the compressed spring try to move them apart. So we can say balls repel each other when they get too close to each other.

If you release the balls (click play) the balls will come back to their equilibrium positions (dotted lines).

This means balls do not experience any force (or balls are in equilibrium) when spring is neither stretched nor compressed.

A ball - spring model for a solid object

Since a ball spring model behaves like a two atom system. Therefore we can create a simple model of a solid object by joining several ball spring systems. As  solid object contains many atoms not just two.

ForceAndMotion_40.gif

The lattice shown in the figure is the simplest kind of crystal but it has all the features we need to understand a rigid body.

This model indicates that, all “rigid” bodies are to some extent elastic, which means they will exert force if we try to change their dimensions slightly by pulling, pushing, twisting, or compressing them.

The Normal Force

When a block of mass m is kept on a table or floor, the gravitational force ForceAndMotion_41.gif pulls it down, but the block remains stationary.

This means the net force acting on the block is zero.

From where the balancing force is coming?

Analysis of Normal force

Two forces ForceAndMotion_43.gif and ForceAndMotion_44.gif are acting on the block, therefore net force ForceAndMotion_45.gif on the block is.

ForceAndMotion_46.gif

Free body diagram of the block is shown in the figure.

ForceAndMotion_47.gif

Now suppose the table and block are moving vertically with an acceleration ForceAndMotion_48.gif.

According to Newton's second law

ForceAndMotion_49.gif

By substituting the value of ForceAndMotion_50.gif and moving it to RHS we get magnitude ForceAndMotion_51.gif of normal force

ForceAndMotion_52.gif

Weight

The weight W of a body is the magnitude of the force required to prevent the body from falling freely on the ground.

ForceAndMotion_53.gif ForceAndMotion_54.gif

How many forces are acting on the object (cantaloupe)?

Two forces are acting on the body: a downward gravitational force ForceAndMotion_55.gif and a balancing upward normal force ForceAndMotion_56.gif of magnitude W.

Consider a body that has vertical acceleration ForceAndMotion_57.gif relative to the ground, which we again assume to be an inertial frame.

We can write Newton's second law for a vertical y axis, with the positive direction upward, as

ForceAndMotion_58.gif

ForceAndMotion_59.gif

Therefore magnitude of normal force ForceAndMotion_60.gif is preventing the object from falling.

Therefore magnitude W of normal force ForceAndMotion_61.gif is called the weight of the object (cantaloupe) and is given as

ForceAndMotion_62.gif

When vertical acceleration of an object is zero, the weight W of the object is equal to the magnitude Fg of the gravitational force on the body.

Apparent weight

What if the vertical acceleration ForceAndMotion_63.gif relative to the ground?

ForceAndMotion_64.gif ForceAndMotion_65.gif

This can be true if scale is hanging in an accelerating elevator.

If the cantaloupe and scale are accelerating vertically say in an elevator with ForceAndMotion_66.gif acceleration, according to Newton's law

ForceAndMotion_67.gif

ForceAndMotion_68.gif

Therefore weight W (magnitude of ForceAndMotion_69.gif) is

ForceAndMotion_70.gif

ForceAndMotion_71.gif is positive if elevator is accelerating upward.

ForceAndMotion_72.gif is negative if accelerating downwards.

Apparent weight of an object is always equal to the normal force acting on the object.

If the cantaloupe and scale are not accelerating vertically ForceAndMotion_73.gif, the apparent weight of the block will be equal to its real weight.

ForceAndMotion_74.gif

Checkpoint - 3

Is the magnitude of the normal force  greater than, less than, or equal to mg if the block and table are in an elevator moving upward (a) at constant speed and (b) at increasing speed?

ForceAndMotion_75.gif

Tension Force

When a block of mass m is hanging on a cord (or a rope, cable, or other such object), the gravitational force ForceAndMotion_76.gif pulls it down, but the block remains stationary.

This means the net force acting on the block is zero.

From where the balancing force is coming?

Characteristics of Tension Force :

Tension has the following characteristics:

It is always directed along the rope

It is always pulling the object

It has the same value along the rope.(for example between points A and B)

ForceAndMotion_78.gif

A cord is often assumed to be mass less (meaning its mass is negligible compared to the body's mass) and un-stretchable.

Some points to be remembered about tension force

Even if the cord runs around a pulley, It pulls on both bodies with the same force magnitude of ForceAndMotion_79.gif.

ForceAndMotion_80.gif ForceAndMotion_81.gif ForceAndMotion_82.gif

The direction of the tension force pulling the object, always remains along the cord, even if it has gone around the pulley.

Force of tension remains even if the bodies are accelerating.

If the cord wraps halfway around a pulley, the net force on the pulley from the cord has the magnitude 2T.

Checkpoint - 4

The suspended body in the figure, weighs 75 N. Is T equal to, greater than, or less than 75 N when the body is moving upward (a) at constant speed, (b) at increasing speed, and (c) at decreasing speed?

ForceAndMotion_83.gif

Recipe for the application of Newton' s law of motion.

Choose the system to be studied.

Make a simple sketch of the system, Fig. (a)

Identify all the forces that act on the system. Label them on the diagram.  

Draw the free body diagram with all the external forces acting on the system to be studied, Fig. (b)

Choose a convenient coordinate system.

Compute components of the forces not parallel to any axis, Fig. (c).

Calculate components of the net force.

Apply Newton’s  second law of motion to the system.

ForceAndMotion_84.gif ForceAndMotion_85.gif ForceAndMotion_86.gif
(a) (b) (c)

Newton’s Third Law

Consider a book and a crate in the figure.

ForceAndMotion_87.gif

If the book exert a force ForceAndMotion_88.gif on the crate, then according to Newton's third law the crate will exert a force ForceAndMotion_89.gif on the book such that

ForceAndMotion_90.gif

Where minus sign means two forces are in opposite direction. We can call the forces between two bodies a third-law force pair.

Newton’s Third Law: When two bodies interact, the forces on the bodies from each other are always equal in magnitude and opposite in direction.

Checkpoint 5

Suppose that the cantaloupe and table in the figure are in an elevator cab that begins to accelerate upward. (a) Do the magnitudes of  ForceAndMotion_91.gif and ForceAndMotion_92.gif increase, decrease, or stay the same? (b) Are those two forces still equal in magnitude and opposite in direction? (c) Do the magnitudes of ForceAndMotion_93.gif and ForceAndMotion_94.gif increase, decrease, or stay the same? (d) Are those two forces still equal in magnitude and opposite in direction?

ForceAndMotion_95.gif ForceAndMotion_96.gif ForceAndMotion_97.gif

Uniform Circular Motion

We know that when a body moves in a circle (or a circular arc) of radius R at constant speed v, it is said to be in uniform circular motion.

The body has a centripetal acceleration ForceAndMotion_98.gif of constant magnitude given by

ForceAndMotion_99.gif

What is the direction of ForceAndMotion_100.gif ?

If we look at the following simulation, ForceAndMotion_101.gif is always directed toward the center of the circle.

Any object moving along a circle, always has centripetal acceleration directed towards its center.

Checkpoint 6

When you ride in a Ferris wheel at constant speed, what are the directions of your acceleration ForceAndMotion_103.gif and the normal force ForceAndMotion_104.gif on you (from the always upright seat) as you pass through
(a) the highest point and
(b) the lowest point of the ride?

ForceAndMotion_105.gif

Problem - 2

In the figure, a car is driven at constant speed over a circular hill and then into a circular valley with the same radius. At the top of the hill, the normal force on the driver from the car seat is 0. The driver’s mass is 70.0 kg.

ForceAndMotion_106.gif

What is the magnitude of the normal force on the driver from the seat when the car passes through the bottom of the valley?

Solution :

At the top of the Hill, net force is ForceAndMotion_107.gif

At the bottom of the Hill, net force is ForceAndMotion_109.gif

Friction Force

When we slide or try to slide an object over a surface, the motion is resisted by a bonding between the object and the surface.

The resistance is called frictional force or simply friction

Static frictional force

When we exert force in an attempt to pull a block to the left, a frictional force is directed to the right.

ForceAndMotion_111.gif

This force exactly cancels our force and the block does not move.

This frictional force is called the static frictional force ForceAndMotion_112.gif.

Now as we increase the magnitude of the applied force, the magnitude of the static frictional force ForceAndMotion_113.gif also increases and the block remains at rest.

Kinetic frictional force

But when the applied force reaches a certain magnitude, however, the block “breaks away” from its intimate contact with the surface and accelerates leftward.

ForceAndMotion_114.gif

The friction force is still present, and opposes the motion.

This frictional force ForceAndMotion_115.gif which opposes the motion is called the kinetic frictional force.

Magnitude of Friction force

When we apply a force ForceAndMotion_116.gif to the block and it does not move, what will be its acceleration?

ForceAndMotion_117.gif

But when the magnitude of applied force reaches a certain maximum value, the block “breaks away” and start moving.

This is the maximum static force of friction ForceAndMotion_118.gif the block can produce.

Experimentally magnitude of maximum static force of friction ForceAndMotion_119.gif is found to be proportional to the magnitude of normal force ForceAndMotion_120.gif

ForceAndMotion_121.gif

where ForceAndMotion_122.gif is the coefficient of static friction and  ForceAndMotion_123.gif is the magnitude of the normal force on the body from the surface.

When the magnitude of the applied force ForceAndMotion_124.gif exceeds ForceAndMotion_125.gif, the body begins to slide along the surface.

When the body begins to slide along the surface or body is in motion, the magnitude of the frictional force rapidly decreases and is called the kinetic frictional force ForceAndMotion_126.gif.

Experimentally magnitude of kinetic frictional force ForceAndMotion_127.gif is found to be proportional to the magnitude of normal force ForceAndMotion_128.gif.

ForceAndMotion_129.gif

where ForceAndMotion_130.gif is the coefficient of kinetic friction.

Magnitude of kinetic frictional force  ForceAndMotion_131.gif is always less than the magnitude of maximum static force of friction ForceAndMotion_132.gif.

Checkpoint 7

A block lies on a floor.
(a) What is the magnitude of the frictional force on it from the floor?
(b) If a horizontal force of 5 N is now applied to the block, but the block does not move, what is the magnitude of the frictional force on it?
(c) If the maximum value  of the static frictional force ForceAndMotion_133.gif on the block is 10 N, will the block move if the magnitude of the horizontally applied force is 8 N?
(d) If it is 12 N?
(e) What is the magnitude of the frictional force in part (c)?  

Problem - 3

In the figure, blocks A and B have weights of 44 N and 22 N, respectively.  

ForceAndMotion_134.gif

Determine the minimum weight of block C to keep A from sliding if ForceAndMotion_135.gif between A and the table is 0.20.

Block C suddenly is lifted off A. What is the acceleration of block A if ForceAndMotion_136.gif between A and the table is 0.15?

Solution :

When Block B is not sliding, its acceleration is zero. Consider forces on Block B when it is not sliding:

When Block A and C are not sliding, their acceleration is zero. Consider horizontal forces on Block A & C when it is not sliding:

When Block C is suddenly taken off, Block A will slide with acceleration a. Now net horizontal forces on Block A will be equal to  its mass times acceleration:

Problem - 4

A puck of mass m=1.50kg slides in a circle of radius r=20.0cm on a frictionless table while attached to a hanging cylinder of mass m=2.50kg by a cord through a hole in the table (See Figure).   

ForceAndMotion_143.gif

What speed keeps the cylinder at rest?

Solution :

Suppose the puck is rotating with constant speed v in a circle of radius r=20.0cm.

The tension force T will always act perpendicular to the direction of the velocity of puck.

Net force on the puck

ForceAndMotion_144.gif ForceAndMotion_145.gif

When we consider only the Hanging Cylinder as a system, the net force is in a circle of radius r=20.0cm

ForceAndMotion_146.gif

Problem - 5

In the following figure, a 1.34 kg ball is connected by means of two mass-less strings, each of length L=1.70m, to a vertical, to a rotating rod. The strings are tied to the rod with separation d=1.70m and are taut. The tension in the upper string is 35 N.    

ForceAndMotion_151.gif

What is the tension in the lower string?

What is the magnitude of the net force ForceAndMotion_152.gif on the ball?

What is the speed of the ball?

What is the direction of ForceAndMotion_153.gif?