Category: Physics

Spry Springs

This is a challenge about springs.

Say we have a few springs. They each have their nominal lengths and their spring constants. We wrap them all around a rail, with sliding spacers in between. In this way we have created a larger spring. We are challenged to find a way to compute the characteristics of this spring–importantly, the spring constant–in a general way. This should be a short, fun activity! But really, it’s actually a surprisingly interesting and challenging task. For a simpler task, try only the first example below. To practice your math after finding a formula, try all of the examples below!

Spring ConstantsNominal Lengths
2N/m, 5N/m2m, 1m
3N/m, 4N/m, 3N/m3m, .5m, 4.5m
1N/m, 2N/m, 8N/m4m, 1m, 3m
.5N/m, 1N/m, 1N/m2m, 3m, 10m
solution

Centrifuge System

Physics is required to solve the first part of this challenge.

Free body diagram illustrating setup for challenge.

The challenge begins with massless pivot anchored to the ground. Connected to the pivot by two unbending metal rods a fixed third rotation apart are a weight and another massless pivot. The weight is 5 cm from the centre, the other pivot, 3 cm. The pivot is clockwise of the weight. Attached to the second pivot, there is also another wight, attached by another unbending rod, this one of length 3 cm. The first weight weighs 5 kilograms, the second, three. A force of 1 newton is applied for 1 second to the first weight, perpendicular to the metal rod, to make the system rotate counterclockwise. The challenge is to determine how many rotations per minute it will spin at.

solution

Captivating Circles BONUS

This bonus science challenge pays tribute to our old Captivating Circles series. It involves physics. You may ask, ‘will there be a Fascinating Frequencies bonus math challenge?’ Yes, 27 weeks after the last problem. Why 27? It’s about 6 months.

Illustration of Cylinders before they begin to roll.

In this bonus challenge, we have a cylinder. It has a smaller cylinder cut out of it. Inside this hole is an even smaller cylinder. The ratio of radii of the big cylinder to that of the hole to that of the small cylinder is 3:2:1, and the smallest one is 0.1 metres. Both the cylinders have equal density. The kinetic friction between the big cylinder and the ground is a kilogram per second. This unit seems weird because we haven’t multiplied by the velocity. That between the small and big cylinders is 5 kg/s. The ground is inclined by 10 degrees in the direction the cylinder is facing. It will begin to roll, and reach a stable speed eventually. The challenge is to figure out how fast.

solution

Unbalanced Balance

Physics is needed for this challenge.

Depiction of scene for Unbalanced Balance.

There is a lever. On one side is a bucket with negligible weight. On the other is a 10 kilogram weight. The bucket is a metre to the left of the pivot, the weight, 2 to the right. Lever has an additional 50 centimetres of unoccupied space. The weight of the lever is 1 kilogram per metre. The weight is attached to a string, and the string goes around a pulley. On the other side is a 7 kilogram weight. The challenge is to figure out how much water at 20° celsius must be poured into the bucket to balance the scale.

solution

Fascinating Frequencies 6

For our final Fascinating Frequencies challenge, we will be using chemistry, physics, simulations, and orbits.

An experiment is held in space station orbiting earth from a geostationary orbit. Unfortunately, the scientists onboard forgot to properly secure one side of the space station. The side pointing prograde. Well, at least not retrograde, but their orbit is still messed up seriously. The reaction was between sodium and fluorine, with 19 kilograms fluorine and 23 kilograms sodium. The mass of the space station is 500 megagrams, including remaining fuel. How many grams of fuel must the scientists burn minimum to bring their rocket back to geostationary orbit? The scientists’ rocket fuses hydrogen and oxygen.

solution

Fascinating Frequencies 5

You will need to use physics for this Fascinating Frequencies challenge.

An interesting device can be created when you have a wheel of gears. Of course, this wheel will have an even number of gears. If the device is anchored and you spin one gear, the others will all also spin at the same frequencies. But if you anchor the gear you spin, the others can spin around it. Try building this device yourself, it’s very cool. Say you make one of these devices with six gears. The radius of each gear is 10 centimetres, and the centripetal force felt by the gear opposite to the anchored one you are spinning is 1 metre per second squared. The challenge is to find what frequency the gear is spinning.

solution

Fascinating Frequencies 4

A physics simulation is needed to solve this challenge.

Diagram of Baffling Bungee Jump.

Bob the Bungee jumper from Baffling Bungee is upset about his bungee jumping experience. If you attempt the problem, you will soon see that Bob’s bungee jump will take too long. Far too long. This is because C5H8 is very springy. In fact, Bob’s bungee jump would take about 2 months … if you were thinking of a mathematical model. You see, the diagram shows a cliff behind Bob. The rubber is so springy he would bash into it, cutting his jump short. But set that problem aside for now. This challenge does not concern those. In bob’s jumps, the ups and downs would gradually, very gradually, become smaller and smaller. But they would still occur at the same frequency. Can you guess today’s challenge? Calculate that frequency.

solution

Fascinating Frequencies 2

This challenge involves orbits.

A geostationary orbit is one where the satellite stays in the same position relative to the ground. This works because the earth spins; the velocity that keeps it over the same spot increases while the velocity needed to orbit decreases. At some point, they intersect. All geostationary orbits must be at the equator, and have the same altitude althroughout. If you wanted to get a satellite into geostationary orbit around a planet with mass of 5,000 yottagrams that completes 1 revolution every 20 hours, how fast would your orbital speed be at apoapsis? The apoapsis is the highest point in an orbit.

solution

Fascinating Frequencies 1

This challenge is the first of a new science challenge series, Fascinating Frequencies. You will need to use physics to solve this challenge.

Some radioactive particles are decaying. They are decaying in different ways, but each type of decay releases the same amount of energy. If the particle undergoes beta decay, it will release an electron and positron, both traveling at half the speed of light in a vacuum. If it undergoes gamma decay, it will release a photon. The challenge is to find the frequency of this photon.

solution

Bouncy Ball

Physics simulations are highly reccomended to solve this science challenge.

You release a bouncy ball from 1 metre above the Earth. Neglect decrease from surface gravity. It will fall and bounce, then fall again, and so on.With each bounce, the ball’s velocity after the bounce is the negative root of it’s speed before it. For example, if it’s velocity was 4 m/s down, it’s velocity would become -2 m/s down, or 2 m/s up. Clearly, this is not a normal bouncy ball. In fact, it breaks several laws of physics. But still, calculate the velocity of the ball after 100 jumps.