Magnetic Linear Accelerator

Have you ever seen one of those roller coasters that shoots out of the station at an insanely high speed? These roller coasters don’t need to climb hills first to use gravitational potential energy—their power comes from magnetism and energy conservation.

A series of electromagnets (magnets made by pumping electrical current through coils of wire) alternately push and pull on the rollercoaster, pumping up its speed pretty quickly. Some engineers have imagined using the same idea to launch objects into space (from, say, a base on the Moon) without using rockets.
In this project, you’re going to build a very simple magnetic accelerator to launch steel balls at targets. What could possibly go wrong?

Problem

Build a simple magnetic linear accelerator.

Materials

• Wooden ruler with groove along the middle
• Four small, powerful magnets (e.g., neodymium magnets)
• Nine steel balls, roughly 5/8” in diameter
• Tape
• Hobby knife
• Safety Goggles

 Procedure

1. Place the ruler (flat side down) on a table.

2. Lay one magnet in the ruler groove, about 2.5” from the ruler’s end. Use the tape to secure the magnet to the ruler and the knife to trim the tape to the size of the magnet.

3. Repeat step 2 with each of the remaining three magnets, placing each about 2.5” away from the preceding magnet.
4. To the right side of each magnet, place two steel balls in the groove.

5. Place a “target” a few inches to the right of the ruler. Your tape dispenser will work fine.

 

6. Place the ninth ball in the groove on the far left end of the ruler (opposite the target).

 

7. Put your safety goggles on.
8. Let the ball go and stand back!

Results

The ball will be attracted to the first magnet and set off a chain reaction of balls firing between the magnets until the last one flies off the ruler at high speed to strike its target.

Why?

What you just saw is a fantastic example of energy conservation. Energy from one ball gets transferred to the next, and then to the next, and so on. But where is all the energy in the last ball coming from if the first ball starts off from rest?
The answer is in the magnets.
Before the starting ball is released, there is potential energy stored up between the ball and the first magnet. The magnet and ball feel an attractive force, but your finger is preventing anything from happening. Once you let go of the ball, it gets drawn towards the magnet (which won’t move because it’s taped down). Potential energy gets converted to kinetic energy—the energy of motion. This is no different then holding a ball in the air and letting it go.
Eventually the ball strikes the magnet—but where does all that energy go? Well, it gets transferred to the balls on the other side of the magnet. The ball closest to the magnet is held pretty tight, but the second ball is farther away and doesn’t feel as strong an attraction to the magnet. This means there’s enough kinetic energy from the first ball to send this ball flying off with nearly the same amount of energy. (That’s why we need two balls stuck to the other side of the magnet: to lessen the attractive force a bit. Try getting it to work with only one ball loaded up next to each magnet and see what happens.)
This second ball is launched at roughly the same velocity as the first ball achieved. As this second ball gets drawn to the second magnet, the attractive force causes it to accelerate and hit the second magnet at a higher velocity than the first ball hit the first magnet. The third ball takes off with the highest velocity achieved by the second ball, and since it gets accelerated by the third magnet in turn, it strikes third magnet faster and harder than the first two balls struck their respective magnets.
Are you seeing a pattern begin to emerge? With each added magnet, more kinetic energy accumulates in each launched ball. The last ball takes off with the combined kinetic energies of all the balls that came before it!
In principle, you can add more rulers and magnets and get the final ball moving as fast as you like—up to a point. Eventually, the balls would be moving fast enough to break the magnets, a limit for which I’m sure your target is very thankful.

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Centripetal and Centrifugal Force

Have you ever wondered why you don’t fall out of an upside down loop on a roller coaster, or why a satellite can stay in orbit around the earth? Centripetal force is a force that causes an object to move along a curved path by pulling the object towards the center of the path. The velocity (speed and direction) of the object is constantly changing because the direction of the object is constantly changing, even though the speed remains the same unless acted upon by an outside force. The direction of such an object at any given point is always perpendicular to the centripetal force.
For a circle, the centripetal force is given by the following equation:
F = ma_c = (mv² / r)
Where F is the force in Newtons, m is the mass of an object in kilograms, and a_c is the centripetal acceleration which can also be described by v²/r, the square of the velocity divided by the radius (the distance from the center of the circle).
For objects traveling in a vertical orientation, meaning at some point they are upside down, the centripetal force must be at least equal to the gravitational force, so the object (or person!) does not fall.

Objective:

Observe centripetal force in action, and use the centripetal force equation to predict the results of the experiment.

Hypothesis:

What will happen to the water in the bucket when the bucket is spun faster? Slower?

Materials:

• Plastic bucket with handle
• Scale
• Large jug
• Water
• Meter stick
• Notebook and pencil

Procedure

1. Measure your arm from the shoulder to your hand. When spinning the bucket, this will be the radius of the circle. Record the length in meters.
2. Weigh the bucket on the scale. Record the weight.
3. Place the large jug on the scale and record the weight.
4. Pour water into the jug and record the weight. Subtract the weight of the jug to get the weight of the water alone.
5. Convert the weight of the water into kilograms. Why is it important that the mass is in kilograms?
6. Pour the water into the bucket.
7. Go outside to an area where it is okay to spill water. With your arm fully extended, swing the bucket around in circles.
8. Swing the bucket slower and slower until the water spills out.
9. Using the centripetal force equation, calculate the velocity of your spin for the mass of water in the bucket. How do you solve for velocity? What is significant about this force? This velocity?
10. Repeat the experiment with different masses of water, or even different radii by tying a rope to the bucket handle.
11. Compare centripetal force to gravity exerted on the water. How much water can you swing for a given velocity?

Results:

The water will spill out of the bucket when the gravitational force of the water exceeds the centripetal force exerted on the water when it is upside down.

Why?

Centripetal force exerted on a spinning object like our bucket of water also leads to an equal and opposite centrifugal force, an apparent force that draws a rotating object away from the center of rotation (thus holding the water in the bottom of the bucket as it passes overhead). Centrifugal force is a consequence of inertia—the tendency of a moving object to want to continue moving in a straight line. As we fling our bucket of water in an arc over our head, the water wants to continue traveling in a straight line, but our string constantly redirects the water so it travels in an arc instead! Water’s inertia resists this redirection, leading to the apparent force that “pulls” the water into the bottom of the bucket. It’s a great example of Newton’s third law: The string pulls on the water to change its direction from a straight line to an arc (centripetal force), and the water’s inertia pulls back (centrifugal force)!
Here’s an analogous situation: Imagine you’re riding as a passenger in your dad’s car. He makes a really sharp turn, and as a result, you feel like you’re being thrown against the inside of the car door. What’s really happening is that your body wants to continue moving forward, but the turning car pulls your body in a new direction. Your body’s intertia resists this pull, because like all objects, it wants to continue traveling in a straight line.
Now, let’s take a look at the math.
To solve for velocity of your swinging bucket, you have to calculate the gravitational force that acts on the water:
F_g=mg
Where F_g is the gravitation force in Newtons, m is the mass of the water and g is the acceleration due to gravity, which is 9.81m/s on Earth.
The water will spill from the bucket when the gravitational force is slightly greater than the centripetal (or centrifugal) force, so for simplicity they can be set to equal each other, the variables rearranged, and solved. It is important that weight (mass) is measured in kilograms because that the units in the equation must be consistent for the equation to be true.

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