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Springs

Springs are very useful. We normally think of a spring as a coil of stiff metal wire. The shock absorbers on your car are surrounded by very strong (and thick) springs. A mechanical watch has a spiral spring to measure time. While it is not a coil spring, a bow acts like a spring to store energy to launch an arrow. Highway bridges are constructed to act like a spring to adjust (slightly) when a heavy load like a truck travels over them.Even you bed (the 'box spring' part) has springs to make the bed comfortable.

A simple application of a spring is the scales in the produce section. The more apples you put in the basket, the more the spring stretches and the more mass the pointer indicates.

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Springs – Hooke's Law

Springs are very useful. When a spring is compressed or stretched, the force the spring exerts is proportional to the distance the spring is displaced from its uncompressed or unstretched position. This is known as Hooke's law:

F = -k x

Where F is the force, x is the displacement from the spring's equilibrium position and k is a constant for that particular spring. The units for the spring constant, k, are N/m. Each spring has it own spring constant.

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Translational Equilibrium

When an object is stationary, the acceleration must be zero because the sum of all forces acting on the object must be zero. We say the object is in equilibrium.

Consider standing on two bathroom scales.The sum of the two are your weight. If youlean to one side the readings are different the two numbers add up to the same in each case.

If you consider all the forces, part of your weight pushes down on each scale. The scales push back on you (you are in equilibrium). In turn, the scales push down on the floor and the floor pushes up on the scales which are in equilibrium.

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Translational EquilibriumThe forces don't have to be in the same direction as with the scales. Consider a sign hanging from two wires. The sign is in equilibrium. The weight of the sign is down. The wires pull at 45o with respect to the horizontal so there are both vertical and horizontal components for each wire supporting the sign. However the three forces acting on the sign add up as vectors to zero since the sign is in equilibrium.

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Rotational EquilibriumAn object can be in rotational equilibrium. Consider the two children on the seesaw. The weight of each is 200N and each is 1 m from the pivot. The girl's weight give a torque of +200 Nm (counter clockwise rotation) and the boy's weight applies a torque of -200 Nm since it tend to rotate the seesaw clockwise. The net torque is then zero and the seesaw is in equilibrium.

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Rotational Equilibrium

If the girl has a weight of 400 N, then for the seesaw to be in equilibrium (Στ = 0), the girl must be at 0.5 m to give a torque of 200 Nm to balance the -200 Nm torque of the smaller boy.The seesaw

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Dynamic Equilibrium

An object can also be in equilibrium if it is moving with constant velocity. The object has zero acceleration since its velocity is not changing and the net force on the object is zero.

Consider sliding a box across the floor at constant velocity.The friction balances your push in the horizontal direction so the net force in the horizontal direction is zero (ΣFx = 0)

Similarly, If the angular velocity, ω, is constant, the object can be in rotational equilibrium and Στ = 0.

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Bouncing Balls

A ball is like a spherical spring. Squeeze a ball a little and it deforms a small amount. Squeeze it more and it deforms more. Each ball is different. Some deform more when you squeeze them than other. Some balls bounce better than others. We can quantify this with the 'coefficient of restitution':

Coefficient of restitution =

This ratio can never be greater than one.

Another measure of a ball's 'bounce' is the ratio of the out going kinetic energy (rebound) over the incoming kinetic energy (collision). This number is also never greater than one because of energy conservation.

Outgoing speed of ballIncoming speed of ball

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Bouncing Balls

If we look at a table of the coefficient of restitution and the rebound KE over the collision KE, you notice that the KE ratio is the square of the coefficient of restitution.

This is what we expect since kinetic energy is ½mv2.

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Bouncing Balls

The 'hardness' of the surface the ball bounces on also has an effect. If you drop a golf ball on a concrete floor it bounces better than on a mattress. The surface that the ball bounces on can also act as a spring. A good tennis racket will absorbs energy and release it back to the ball on the rebound.

You would also expect the ball to rebound faster if the 'surface' from which it rebounds is moving like with a tennis ball striking a moving racket. We should re-do our definition to account for a moving 'surface' as well as the ball moving:

Coefficient of restitution = Relative speed of separation

Relative speed of approachA baseball moving at 30 m/s approaching a bat moving at 20 m/s is approaching the bat at 50 m/s

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Collision of a bat and a baseball

More complex things can happen in the collision of a ball and a bat. If the ball does not strike the ball at the center of mass, the ball and bats will experience a torque and rotate. The bat is not perfectly rigid. The bat can vibrate. All real (not perfectly rigid) object have 'vibrational modes'. These vibrational modes can be complex and depend on the geometry and distribution of the mass. For a baseball bat, a good design has a 'node' at the center of mass. A mode is a position where there is no vibrational motion. Bridge engineers have to pay particular attention to vibrational modes.

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Uniform Circular MotionGoing around a curve on a road or riding on a carousel are examples of uniform circular motion i.e., constant speed going in a circle.

The time it takes to go around one complete circle is the period:

Period = T = 2πrv

By definition the speed is constant. The velocity has magnitude v but is tangent to the circle and constantly changing direction.

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Uniform Circular Motion

Since the velocity is changing, there must be an acceleration. The circular (or centripetal) acceleration is given by:

acceleration circular

= ac =

What about the direction of the acceleration? The acceleration always points towards the center of the circle. It constantly changes the velocity vector back towards the center of the circle to keep the object moving in a circle.

v2

r

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Roller Coasters

A roller coaster provide good examples of work, energy, gravity and uniform circular motion. To get things going, a chain drive does work on the car. This work gives the car gravitational potential energy to the highest point on the track. At this point gravity takes over and energy is converted to kinetic energy. Some energy may be dissipated as heat by friction. The car goes fast down the hills (GPE KE) and then slows (KE GPE) going up the hill.

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Roller CoasterNear the bottom is a 'loop the loop'. The track applies force so the car starts up the side of the loop. The car slows down (KE GPE). You feel the push toward the center of the loop. Gravity still pulls you down. As you go over the top of the loop (and if you have enough KE left) the only force acting on you is gravity. At this point, gravity is providing the centripetal force. You and the car are weightless.