Category: Science

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

Fascinating Frequencies BONUS

This challenge has calculus and algebra.

We begin this problem by defining a few variables and functions, see below.
Let a ‘functional triangle’ be a relationship between three functions shown, denoted by the two large circles.
The properties of such are shown.
Then, a ‘meta-functional triangle’ is denoted in the same manner be a functional triangle with the given additional property.
The challenge is to find a meta-functional triangle. Good luck!

\text{let }x,\,y,\,z\in\R,\;x\leq y\leq z\\[8pt] \text{if }\;\exists!\;\{x,\,y,\,z\}: \operatorname fa=\operatorname gb=\operatorname hc\;\; \forall\;\;\{a,\,b,\,c\}\equiv\{x,\,y,\,z\},\\[4pt] \text{then }\;f\bigcirc g\bigcirc h\\[8pt] \text{if }\ f\bigcirc g\bigcirc h\;\ \text{and }\ f’\circledcirc g’\circledcirc h’,\ \;\text{then}\;f\circledcirc g\circledcirc h
SOLUTION

Esoteric Elastics

This challenge is about elastics.

We begin with 2 elastics. Elastic A has nominal length 3 centimetres and spring constant 2 kilograms per square second. This notation seems unfamiliar but remember that we multiply by the displacement, in metres, and get a force, in mkg/s2. Elastic B has nominal length 5 centimetres and spring constant 3 kg/s2. The long elastic it threaded through the small one. The 2 ends of the long elastic are folded together. 2 identical rods are threaded through them. They are then pulled apart. The elastics are held in place at equal heights on the rods so they cannot move up or down. They are at equal heights and can rotate frictionlessly. The rods are then fastened to the surface of the earth some distance apart. The rods do not sway, bend, or in any way stop being vertical. A weight is placed halfway between the 2 rods, resting on the elastics. It weighs 1 gram. The weight does not cause any friction on the elastics. Elastics A and B are stretched to equal lengths. How far apart are the rods?

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 3

Some chemistry principles are required to solve this challenge.

Diagram of pond after being hit by rock. waves are just for illustration purposes and are not part of the problem.

In this Fascinating Frequencies challenge, we will cover fluid dynamics. A perfect homogeneous sphere with radius 10 cm and mass 1 kg hits the middle of a pond of pure water on earth. It’s velocity is 1 m/s downward. The temperature of the pond is a uniform 20° celsius. The pond will begin to ripple. There are no animals in the pond to interfere with this. The challenge is to calculate the frequency of the ripples.

solution