Conservation of Energy with Springs AP style

Conservation of Energy with Springs Example One: A 6 x 105 kg subway train is brought to a stop from a speed of 0.5 m/s in 0.4 m by a large spring bumper at the end of its track. What is the force constant k of the spring? Assume a linear spring unless told otherwise! Solution: All the initial kinetic energy of the subway car is converted into elastic potential energy. ½ mv2 = ½ kx2 Here x is the distance the spring bumper is compressed, x = 0.4 m.

K = ½ (0.25 kg) v2 = 2.5 J = Us = Ws v2 = 20 m2 / s2 v = 4.47 m/s A spring with spring constant k = 500 N/m is compressed a distance x = 0.10 m. A block of mass m = 0.250 kg is placed next to the spring, sitting on a frictionless, horizontal plane. When the spring and block are released, how fast is the block moving as it leaves the spring? When the spring is compressed, work is required and the spring gains potential energy, Us = ½ k x2 Us = ½ (500 N/m) (0.10 m)2 Us = 2.5 J As the spring and mass are released, this amount of work done to the mass to change its kinetic energy from zero to a final value of K = ½ m v2 K = ½ (0.25 kg) v2 = 2.5 J = Us = Ws v2 = 20 m2 / s2 v = 4.47 m/s

How fast is the block moving when the spring is compressed 0.05 m? E = 2.5 J 2.5 J = ½ mv2 + ½ kx2

mgh1 + ½ mv12 + ½ kx12 = mgh2 + ½ mv22 + ½ kx22 E1 = E2 mgh1 + ½ mv12 + ½ kx12 = mgh2 + ½ mv22 + ½ kx22 Be careful about measuring height!

What about objects suspended from a spring? There are 3 types of energy involved: Kinetic, gravitational and spring potential However, since potential energy is determined using a reference point, you may choose the reference point for potential energy to be at the equilibrium location (with the mass attached), rather than the position where the spring is unstretched. By doing so, the total potential energy, including both Us and Ug can be shown to be Us + Ug = U = ½ ky2 where y is the displacement measured from the equilibrium point. equilibrium y E = K + U

Mechanical energy will be lost in the form of heat energy. If there is kinetic friction or air resistance, mechanical energy will not be conserved. Mechanical energy will be lost in the form of heat energy. The DIFFERENCE between the original energy and the final energy is the amount of mechanical energy lost due to heat. Final energy – original energy = energy loss

Let’s try one… A 2 kg cannonball is shot straight up from the ground at 18 m/s. It reaches a highest point of 14 m. How much mechanical energy was lost due to the air resistance? g = 10 m/s2 Final energy – original energy = Energy loss mgh – ½ mv2 = Heat loss 2 kg(10 m/s2)(14 m) – ½ (2 kg)(18 m/s)2 = ??

And one more… A 1 kg flying squirrel drops from the top of a tree 5 m tall. Just before he hits the ground, his speed is 9 m/s. How much mechanical energy was lost due to the air resistance? g = 10 m/s2 Final energy – original energy = Energy loss

Don’t even think about it… Sometimes, mechanical energy is actually INCREASED! For example: A bomb sitting on the floor explodes. Initially: Kinetic energy = 0 Gravitational Potential energy = 0 Mechanical Energy = 0 After the explosion, there’s lots of kinetic and gravitational potential energy!! Did we break the laws of the universe and create energy??? Of course not! NO ONE, NO ONE, NO ONE can break the laws! The mechanical energy that now appears came from the chemical potential energy stored within the bomb itself! Don’t even think about it…

Law of Conservation of Energy energy cannot be created or destroyed. According to the Law of Conservation of Energy energy cannot be created or destroyed. But one form of energy may be transformed into another form as conditions change.