Forces and motion

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Topic updated on 10/13/2020 03:27pm

Freefall

A free fall is a downward motion without any initial force or velocity. Our earth has the inherent property of attracting items towards it. Hence a free fall is a natural phenomenon on earth for any object at any height without any support.

Free Fall Speed

An object in motion is governed by two parameters, the velocity and the acceleration. The same concept also applies to free fall. It is too obvious to conclude that an object under free fall has ‘only one’ direction, invariably, vertical downwards. Hence in cases of free fall the velocity can be called in its scalar terms as ‘Free Fall Speed’.

Now we observe that in a free fall, the speed is not uniform. You might have observed that two stones dropped from different heights hit the ground with different velocities. The higher the height, greater is the speed of an object when it reaches the ground. It means that there is some acceleration is imparted on objects under free fall and the same is defined as ‘free fall acceleration’.

Free Fall Acceleration

As per Newton’s second law a force is needed to change the state of rest of any object and hence the law applies to free fall of objects. The action of ‘free fall’ takes place, as we mentioned already, due to an attraction force or gravitation force. When this force is exerted on the object, it is subjected to an acceleration called ‘Free Fall Acceleration’.

This free fall acceleration is constant but naturally the free fall speed is a variable quantity, varies with the time or distance of motion.

In cases of free fall, the distance is better referred as height as the direction is invariably vertical. The free fall acceleration is well known as ‘acceleration due to gravity’. For all practical purposes it is considered as constant at all places of earth and is denoted by the letter ‘g’.

A free fall occurs for an object only due to free fall acceleration and hence its parameters of motion are independent of its mass, when you ignore the air resistance. Normally, the air resistance is very low except in cases of objects with large areas and hence it is a safe bet to ignore that. But at the same time one must realize that the force acquired by an object under free fall is certainly proportional to the mass of the object. In fact such force of the object is nothing but the Net Weight of the object and also its momentum as it hits the ground (mass times the final velocity) depends on the mass of the object.

Let us derive the three important free fall equations considering the free fall without air resistance. Let us assume ‘v’ is the final velocity, ‘h’ is the height and ‘t’ is the time related to a free fall. Obviously the initial velocity for a free fall is 0 and the acceleration in this case is ‘g’.

The final velocity is generally defined as the sum of the initial velocity and the product of acceleration and time.

Therefore, for a free fall,

v = 0 + gt

v = gt ………(1)

The second free fall equation is derived as follows :

The distance covered by an object in any motion is the product of average velocity and time.

Therefore, for a free fall,

h = ½(v + 0)(t) = ½ vt = ½ gt $\times$ t

h = ½gt² ……….(2)

Let us eliminate ‘t’ from these two equations to arrive at the third free fall equation,

v2 = 2gh ……….(3)

Projectile motion

Projectile motion is an example of curved motion with constant acceleration. It is two dimensional motion of a particle thrown obliquely into the air.

Consider the motion and path followed by the ball when it moves in the curved path. We will make two assumptions here:

a) First assumption is that the free fall acceleration (g) remains constant and does not change its value during the motion of the ball.

b) Resistance offered by the ball is negligible.

If we consider the motion and the assumptions stated above, we will find that :

The path of the projectile (ball here) is always a parabola.
The path followed by the projectile is termed as the “trajectory of the projectile“.
Projectile feels only one force while in motion, which is the force of gravity.

Projectile

Projectile Motion Equation

Projectile motion is a two dimensional concept and it follows the two dimensional kinematics. A projectile has both the horizontal and the vertical components of motion.

Projectile motion can be stated as the:

y = $\frac{1}{2}$ (at²) + v₀ t + y₀

Where,
y = height
t = time
a = acceleration of the projectile because of gravity
V₀ = Initial velocity of the projectile
Y₀ = Initial height of the projectile

Horizontal Component of the Velocity : Whenever the projectile is thrown or follows the trajectory, the horizontal component of the velocity does not changes and the displacements covered by the horizontal components of the velocity are uniform. In other words, final horizontal velocity component is equal to the initial velocity component.

Now the point here to note is that when the projectile follows the trajectory, gravity force does not affect or does not make any change in the horizontal velocity component of the velocity.

Vertical Component of the Velocity : Vertical component of the velocity does not remain constant during the projectile motion. Gravity force acts on it and changes the vertical component of the velocity of the projectile. The displacements covered by the vertical component of the velocity are not uniform.

For the vertical component of the velocity during the projectile motion, change in both the magnitude and direction takes place. If the projectile is moving in the upward direction, then the vertical component of the velocity is in the upward direction and decrease in its magnitude takes place.

On the other hand, when the projectile moves in the downward direction, the direction of the vertical components of the velocity is in the downward direction and increase in the magnitude takes place.

Range Equation for Projectile Motion

Equations involving Vertical Motion	Equations involving Horizontal Motion	Explanation of Symbols used

V_(iy) = V_i sin $θ$	V_(ix) = V_icos $θ$	V_i= Magnitude of Initial Velocity V_(iy) = Y component of Initial Velocity V_(ix) = X component of Initial Velocity
V_(fy) = V_(iy) + a_yt	V_(fx) = V_(ix)	V_fy = Y component of Velocity at time ‘t’ V_fx= X component of Velocity at time ‘t’ (note that the X component remains constant) a_y= acceleration in vertical direction, which in the case of projectile motion would be -9.8 m/s²
Y_f – Y_i = V_(iy)t + ½a_yt²	X_f – X_i = V_(ix)t	Y_i = Initial Y co-ordinate of Projectile Y_f = Y co-ordinate of Projectile at time ‘t’ X_i = Initial X co-ordinate of Projectile X_f = X co-ordinate of Projectile at time ‘t’
(V_fy)² = (V_(iy))² + 2a_y(Y_f– Y_i)		Symbols already described above
Y_f – Y_i=t. $\frac{(V_{(i y)} + V_{(f y)})}{2}$		Symbols already described above

Maximum Projectile Range : Expression

Now, lets look at the expression for projectile range using the above formula, Let the projectile start at (0, Yi) co-ordinates with a speed of Vi = v, and angle $θ$ with the horizontal surface. After some time t, it strikes the ground at a distance of Xf. The value of Xf gives the range of the projectile

The figure given below aids the visualization of the motion :

Projectile Range

In this figure, the range of the projectile is given by the formula,

d = $X_{f}$ = $(\frac{(V c o s θ)}{g})$ $(V s i n θ + \sqrt{((V s i n θ)^{2} + 2 g Y_{i})})$

Using the above equation one can make a graph of `theta` versus `d` for different `theta`, and see where the value of `d` maximizes. This will be the value of maximum projectile range. Moreover, this equation reduces to a very simple form when the projectile starts form ground level, that is when $Y_{i}$ = 0.

The equation then becomes :

d =

X_{f}

= $\frac{(2 V^{2} c o s θ s i n θ)}{g}$

d =

X_{f}

= $\frac{(V^{2} s i n (2 θ))}{g}$

Using the above equation we can very easily find the expression for maximum projectile range in this simple situation. We know that the maximum value of sin 2 $θ$ is 1.

Therefore, the maximum range of the projectile is
d = $X_{f}$ = $\frac{V^{2}}{g}$

Also, the value of 2 $θ$ for which sin 2 $θ$ = 1 is $90^{\circ}$ . Therefore, the value of $θ$ = 90/2 = $45^{\circ}$

Horizontal Projectile Motion

This is a type of Projectile motion in which projectile does not follow path in the upward direction or it does not have upward trajectory and the initial velocity of the projectile is also zero. This type of projectile motion is called horizontal projectile motion. This motion generally occurs when the projectile is shot straight without forming any angle with the horizontal surface and the projectile falls downward until it hits the ground.

Exemplary Horizontal Projectile motion is shown in the figure below.

As shown in the figure below, the initial component of the vertical components of the velocity is zero. Horizontal velocity component of the projectile remains constant as the gravity does not affect it. Direction of the vertical component of the velocity is in downward direction during the trajectory. The magnitude of the vertical component of the velocity increases as the projectile moves downward, the force of gravity acts on it, results in acceleration of the projectile.

The figure above illustrates a body thrown horizontally from a point O with a velocity The point O is at a certain height above the ground. Let x and y be the horizontal and vertical distances covered by the projectile, respectively, in time t. Therefore, at time t, the projectile is at p.

In order to calculate x, let us consider the horizontal motion, which is uniform motion. This is because the only force acting on the projectile is the force of gravity. This force acts vertically downwards and hence the horizontal component in zero. Therefore, the equations of motion of the projectile for the horizontal direction is just the equation of uniform motion in a straight line.

$∴$ x = vt —————— (i)

In order to calculate y, the vertical motion of the projectile is considered. Since the vertical motion is controlled by the force of gravity, it is an accelerated motion. The initial velocity, v_y (0), in the vertically downward direction is zero. Since the Y-axis in the figure above is taken downwards, the downward direction is regarded as the positive direction. So, the acceleration of the projectile is + g.

$∴$ from the equation
y(t) = V_y(0)t + ½ a_y t²
We have y(t) = ½ gt² ————-(2)

Here v_y (0) is taken as zero because both distance and time are being measured from the origin O.

From equation (1)
t = $\frac{x}{v}$

Substituting for t from the above equation in equation (2) we have,

y(t) = ½g ${(\frac{x}{v})}^{2}$ = ( $\frac{g}{2 v^{2}}$ )x²

$∴$ y = kx² Where k = $\frac{g}{2 v^{2}}$ ……………..(3)

is a constant for a projectile projected upwards with a definite velocity v and at a place with a definite value of ‘g’.

Equation (3) is a second-degree equation in x, a first-degree equation in y and is the equation of a parabola. Therefore, a body thrown horizontally from a certain height above the ground follows a parabolic trajectory till it hits the ground.

Resultant Velocity of a Horizontal Projectile:

In this section, let us calculate the resultant velocity of the projectile $\vec{V}$ , at any point p on the trajectory, in an interval of time t. V_xand V_y are the horizontal and vertical components of $\vec{V}$ as illustrated in the figure below.

Since, the horizontal motion of the projectile is uniform, ${\vec{V}}_{x}$ = $\vec{V}$

However, the motion in the vertical direction is an acceleration one.

$∴$ V_y(t) = V_y (0) + a_y t

Since O is considered to be the origin, V_y (0) = 0

$∴$ V_v (t) = gt
$∴$ The magnitude of the resultant velocity $\vec{V}$ is given by,

| $\vec{V}$ | = V =√ ( $\sqrt{V_{x}^{2} + V_{y}^{2}}$

$∴$ V = √( $\sqrt{V^{2} + g^{2} t^{2}}$

The direction is given by tan $β$ = $\frac{V_{y}}{V_{x}}$ = $\frac{g t}{V}$

$∴$ $β$ = tan^-1 $\frac{g t}{V}$

Motion -The change of position of a body with time is called motion.

It was Galileo who first realized that motion of a body is independent of its mass, it is the change of state of motion or state of rest.

Inertia

A body is said to possess a property called inertia The inertia of a body is measured by its reluctance to change its state of motion or state of rest. Examples

One’s body movement to the side when a car makes a sharp turn.
Tightening of seat belts in a car when it stops quickly.

Newton’s first law

A body continues its state of rest or moves with uniform velocity unless acted on by some external force.

Linear momentum

Linear momentum is a vector quantity defined as the product of an object’s mass m and its velocity v. Linear momentum is denoted by the letter p .

Note that a body’s momentum is always in the same direction as its velocity vector. The units of momentum are kg m/s.

p=mv

Newton’s second law

The rate of change of momentum of a body is directly proportional to the external force acting on the body and change in momentum takes place in the direction of force.

Newton’s third law

Whenever an object exerts a force on another object,the second object exerts an equal and opposite force on the first.These are called action, reaction. For every action there is always an equal and opposite reaction.

Image result for newton's third law

Note

Forces always appear in pair.
The line of force is always in the same line.
Forces are of same kind.
Forces act on different bodies.

Friction

Friction is the force resisting the relative motion.It comes to existence at the common boundary of two bodies in contact when one of them either moves or tends to move relative to other. Friction acts tangental to the surface and it is directed such that it opposes relative motion.

The surface of a body is never perfectly smooth, little prominences and hollows are always present.When two body is in contact the prominences of one are interlocked with hollows of other.In the relative motion of the bodies these little prominences are deformed and give rise to frictional forces.

Sliding friction

When a body slides over another body , the force that opposes relative motion is called sliding friction.

Specific examples of sliding friction include:

Rubbing both hands together to create heat
A sled sliding across snow or ice
A person sliding down a slide is an example of sliding friction

Rolling friction

When a round object rolls over some surface then rolling friction comes into play.Rolling friction is much smaller than sliding friction.

Examples of Rolling Friction

A car will eventually come to a stop if just allowed to roll as the friction between the road surface and the wheels causes friction that causes the vehicle to stop.
Bike wheels that are thicker will lessen the potential speed of the bike because there is a greater wheel surface to create friction against the surface which will slow the bike.
Heavy duty trucks get greater gas mileage when tread begins to wear on the tires because there is less rolling friction, allowing the truck to move more quickly with less resistance.
A skateboard set on a slight decline will eventually stop itself because of the resistance caused by the friction between the wheels and the surface.

Kinetic friction

When two bodies that are in contact with each other and move rubbing the surfaces that are in contact, the friction existing between them is called kinetic friction. The direction of the force is such that the relative slipping is opposed by the retarding force .

Magnitude of Kinetic Friction
Fk = μk R
Where,
Fk = magnitude of kinetic friction
R = Normal reaction
μk = coefficient of kinetic friction

Note-The coefficient of friction does not depend upon the speed of the sliding bodies. If the surfaces are smooth then it will be small, and it will be large if the surface is rough

Static Friction

The opposing force which comes into play when an object does not move over another object, even when the force is applied to make it move is called Static Friction. Static friction prevents objects from sliding or rolling over each other.
For example, when we push a heavy object, because of the friction the object is unable to move through the surface. Then we apply some force on the object. Once it moves through the surface, it is very easy to move further.
When we were unable to move the object, it was static friction. When it was moving we were overcoming the kinetic friction, which is found to be less than static friction.

Fs = μs R
where,
μ is is the Coefficient of Static friction
R is the Normal reaction

Static friction is greater in magnitude than kinetic friction in any situation.

Relationship between limiting frictional force and normal reaction

Limiting frictional force between two given surfaces depend on the normal reaction between them.To investigate relationship do an experiment with known masses placed on the block.

Initially without mass(m) , apply the force(F) and increase the force until the block in the verge of moving.At that moment take the reading of spring balance.
Now add the known mass(m) to the block and find the force(F) when the block is in the verge of moving.
Do the practical using atleast six different masses(m).

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