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Galileo’s Inclined Planes

This post is based on the maths and ideas of Hahn’s Calculus in Context – which is probably the best mathematics book I’ve read in 20 years of studying and teaching mathematics.  Highly recommended for both students and teachers!

Hahn talks us though the mathematics, experiments and thought process of Galileo as he formulates his momentous theory that in free fall (ignoring air resistance) an object falling for t seconds will fall a distance of ct² where c is a constant.  This is counter-intuitive as we would expect the mass of an object to be an important factor in how far an object falls (i.e that a heavier object would fall faster).  Galileo also helped to overturned Aristotle’s ideas on motion.  Aristotle had argued that any object in motion would eventually stop, Galileo instead argued that with no friction a perfectly spherical ball once started rolling would roll forever.  Galileo’s genius was to combine thought experiments and real data to arrive at results that defy “common sense” – to truly understand the universe humans had to first escape from our limited anthropocentric perspective, and mathematics provided an opportunity to do this.

Inclined Planes

Galileo conducted experiments on inclined planes where he placed balls at different heights and then measured their projectile motion  when they left the ramp, briefly ran past the edge of a flat surface and then fell to the ground.  We can see the set up of one ramp above.  The ball starts at O, and we mark as h this height.  At an arbitrary point P we can see that there are 2 forces acting on the ball, F which is responsible for the ball rolling down the slope, and f, which is a friction force in the opposite direction.  At point P we can mark the downwards force mg acting on the ball.  We can then use some basic rules of parallel lines to note that the angles in triangle PCD are equal to triangle AOB.

Galileo’s times squared law of fall

We have the following equation for the total force acting on the ball at point P:

We also have the following relationship from physics, where m is the mass and a(t) the acceleration:

This therefore gives:

Next we can use trigonometry on triangle PCD to get an equation for F:

and so:

Next we can use another equation from physics which gives us the frictional force on a perfectly spherical, homogenous body rolling down a plane is:

So this gives:

We can then integrate to get velocity (our constant of integration is 0 because the velocity is 0 when t = 0)

and integrate again to get the distance travelled of the ball (again our constant of integration is 0):

When Galileo was conducting his experiments he did not know g, instead he noted that the relationship was of the form;

where c is a constant related to a specific incline.  This is a famous result called the times squared law of fall.  It shows that the distance travelled is independent of the mass and is instead related to the time of motion squared.

Velocity also independent of the angle of incline

Above we have shown that the distance travelled is independent of the mass – but in the equation it is still dependent on the angle of the incline.  We can go further and then show that the velocity of the ball is also independent of the angle of incline, and is only dependent on the height at which the ball starts from.

If we denote as t_b as the time when the ball reaches point A in our triangle we have:

This is equal to the distance from AO, so we can use trigonometry to define:

This can then be rearranged to give:

this is the time taken to travel from O to A.  We can the substitute this into the velocity equation we derived earlier to give the velocity at point A.  This is:

This shows that the velocity of the ball at point A is only dependent on the height and not the angle of incline or mass.  The logical extension of this is that if the angle of incline has no effect on the velocity, that this result would still hold as the angle of incline approaches and then reaches 90 degrees – i.e when the ball is in free fall.

Galileo used a mixture of practical experiments on inclined planes, mathematical calculations and thought experiments to arrive at his truly radical conclusion – the sign of a real genius!

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Time dependent gravity and cosmology!

In our universe we have a gravitational constant – i.e gravity is not dependent on time.  If gravity changed with respect to time then the gravitational force exerted by the Sun on Earth would lessen (or increase) over time with all other factors remaining the same.

Interestingly time-dependent gravity was first explored by Dirac and some physicists have tried to incorporate time dependent gravity into cosmological models.  As yet we have no proof that gravity is not constant, but let’s imagine a university where it is dependent on time.

Inversely time dependent gravity

The standard models for cosmology use G, where G is the gravitational constant.  This fixes the gravitational force as a constant.  However if gravity is inversely proportional to time we could have a relationship such as:

Screen Shot 2020-04-15 at 10.28.17 AM

Where a is a constant.  Let’s look at a very simple model, where we have a piecewise function as below:

Screen Shot 2020-04-15 at 10.28.35 AM

This would create the graph at the top of the page.  This is one (very simplistic) way of explaining the Big Bang.  In the first few moments after t = 0, gravity would be negative and thus repulsive [and close to infinitely strong], which could explain the initial incredible universal expansion before “regular” attractive gravity kicked in (after t = 1).  The Gravitational constant has only been measured to 4 significant figures:

G = 6.674 x 10-11m3kg-1s-2.

Therefore if there is a very small variation over time it is possible that we simply haven’t the accuracy to test this yet.

Universal acceleration with a time dependent gravitational force

Warning: This section is going to touch on some seriously complicated maths – not for the faint hearted!  We’re going to explore whether having a gravitational force which decreases over time still allows us to have an accelerating expansion of the universe.

We can start with the following equation:

Screen Shot 2020-04-15 at 1.44.09 PM

To work through an example:

Screen Shot 2020-04-15 at 1.46.19 PM

This would show that when t = 1 the universe had an expansion scale factor of 2.  Now, based on current data measured by astronomers we have evidence that the universe is both expanding and accelerating in its expansion.  If the universal scale factor is accelerating in expansion that requires that we have:

Screen Shot 2020-04-15 at 1.49.45 PM

Modelling our universe

We’re going to need 4 equations to model what happens when gravity is time dependent rather than just a constant.

Equation 1

Screen Shot 2020-04-15 at 1.50.45 PM

This equation models a relationship between pressure and density in our model universe.  We assume that our universe is homogenous (i.e the same) throughout.

Equation 2

Screen Shot 2020-04-15 at 1.50.54 PM

This is one of the Friedmann equations for governing the expansion of space.  We will take c =1 [i.e we will choose units such that we are in 1 light year etc]

Equation 3

Screen Shot 2020-04-15 at 1.51.00 PM

This is another one of the Friedmann equations for governing the expansion of space.  The original equation has P/(c squared) – but we we simplify again by taking c = 1.

Equation 4

Screen Shot 2020-04-15 at 1.51.06 PM

This is our time dependent version of gravity.

Finding alpha

We can separate variables to solve equation (3).

Screen Shot 2020-04-15 at 1.58.20 PM

Substitution

We can use this result, along with the equations (1) and (4) to substitute into equation (2).

Screen Shot 2020-04-15 at 2.00.23 PM

Our result

Now, remember that if the second differential of r is positive then the universal expansion rate is accelerating.  If Lamba is negative then we will have the second differential of r positive.  However, all our constants G_0, a, B, t, r are greater than 0.  Therefore in order for lamda to be negative we need:

Screen Shot 2020-04-15 at 2.05.57 PM

What this shows is that even in a universe where gravity is time dependent (and decreasing), we would still be able to have an accelerating universe like we see today.  the only factor that determines whether the universal expansion is accelerating is the value of gamma, not our gravity function.

This means that a time dependent gravity function can still gives us a result consistent with our experimental measurements of the universe.

A specific case

Solving the equation for the second differential of r is extremely difficult, so let’s look at a very simple case where we choose some constants to make life as easy as possible:

Screen Shot 2020-04-15 at 2.14.02 PM

Substituting these into our equation (2) gives us:

Screen Shot 2020-04-15 at 2.14.10 PM

We can then solve this to give:

Screen Shot 2020-04-15 at 2.14.19 PM

So, finally we have arrived at our final equation.  This would give us the universal expansion scale factor at time t, for a universe in which gravity follows the the equation G(t) = 1/t.

Screen Shot 2020-04-15 at 2.22.58 PM

For this universe we can then see that when t = 5 for example, we would have a universal expansion scale factor of 28.5.

So, there we go – very complicated maths, way beyond IB level, so don’t worry if you didn’t follow that.  And that’s just a simplified introduction to some of the maths in cosmology!  You can read more about time dependent gravity here (also not for the faint hearted!)

 

 

 

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Projectile Motion III: Varying gravity

We can also do some interesting things with projectile motion if we vary the gravitational pull when we look at projectile motion.  The following graphs are all plotted in parametric form.

Screen Shot 2020-04-06 at 1.17.08 PM

Here t is the parameter, v is the initial velocity which we will keep constant, theta is the angle of launch which we will vary, and g is the gravitational constant which on Earth we will take as 9.81 m/s2.

Earth 

Say we take a projectile and launch it with a velocity of 10 m/s.  When we vary the angle of launch we get the folowing graphs:

Screen Shot 2020-04-08 at 9.56.08 AM

On the y axis we have the vertical height, and on the x axis the horizontal distance.  Therefore we can see that the maximum height that we achieve is around 5m and the maximum horizontal distance is around 10m.

Other planets and universal objects

We have the following values for the gravitational pull of various objects:

Enceladus (Moon of Saturn): 0.111 m/s2, The Moon: 1.62 m/s2,  Jupiter: 24.8 m/s2, The Sun: 274 m/s2, White dwarf black hole surface gravity: 7×1012m/s2.

So for each one let’s see what would happen if we launched a projectile with a velocity of 10 m/s.  Note that the mass of the projectile is not relevant (though it would require more force to achieve the required velocity).

Enceladus:

Screen Shot 2020-04-08 at 10.24.18 AM

The Moon:

Screen Shot 2020-04-08 at 10.25.23 AM

Jupiter:

Screen Shot 2020-04-08 at 10.26.10 AM

The Sun:

Screen Shot 2020-04-08 at 10.26.56 AM

Black hole surface gravity:

This causes some issues graphically!  I’ll use the equations derived in the last post to find the coordinates of the maximum point for a given launch angle theta:

Screen Shot 2020-04-08 at 10.35.01 AM

Here we have v = 10 and g = 7×1012m/s2.  For example if we take our launch angle (theta) as 45 degrees this will give the coordinates of the maximum point at:

(7.14×10-12, 3.57×10-12).

Summary:

We can see how dramatically life would be on each surface!  Whilst on Earth you may be able to throw to a height of around 5m with a launch velocity of 10 m/s., Enceladus  would see you achieve an incredible 450m.  If you were on the surface of the Sun then probably the least of your worries would be how hight to throw an object, nevertheless you’d be struggling to throw it 20cm high.  And as for the gravity at the surface of a black hole you wouldn’t get anywhere close to throwing it a nanometer high (1 billionth of a meter).

 

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