Phys 11: 5.2 Mechanical Energy Mechanical energy is the energy that is possessed by an object due to its motion or due to its position. Mechanical energy can be either kinetic energy (energy of motion) or potential energy (stored energy of position). http://www.physicsclassroom.com/class/energy/Lesson-­‐1/Mechanical-­‐Energy Kinetic energy is the energy associated with the motion of an object. A body at rest will gain kinetic energy if work is done on it by an external force that causes the body to move. If we want the body to be moving at speed v, we can calculate the amount of work that must be done: From last class: W = F Ÿ d But we know from Newton’s second law: F = m a (1) So we get: W = (m a)Ÿd (2) Or: W = m (a Ÿ d) (3) Now, remember from kinematics: vf2 = v02 + 2ad (4) 𝒗𝒇 !
In this case v0 = 0 and solving for ad we get: 𝒂𝒅 = (5) !
Plugging (5) into (3) we find that the work needed to get an object at rest moving at speed v is: 𝒗𝒇 !
!
W = 𝑚 ! 𝑜𝑟 W = ! 𝑚𝑣 ! Since work done on the object to get it to speed v is transferred to the object, we find that the kinetic energy of an object is: !
𝐸! = ! 𝑚𝑣 !
Potential energy is the energy stored in an object. An object will have a certain amount of gravitational potential energy because of its position in a gravitational field. To lift a ball with a mass m to a height h above the floor, the work done is: W = F Ÿ d = F Ÿ h But since the force required is the force to combat gravity (F = mg), we get: W = mgh This is the amount of gravitational potential energy transferred to the ball of mass m lifted to a height h, and so we get the general equation: 𝐸! = 𝑚𝑔ℎ
Note: the height, h, should be measured from a reference point, which is a height where h is considered to be zero. e.g. In the diagram, the table is 1.0 meters tall and the book, with a mass of 1.0 kg, is being lifted a distance of 0.50 meters up from the table top. If the top of the table is the reference point, 𝐸! = 𝑚𝑔ℎ = 1.0 𝑘𝑔 (9.81 𝑚/𝑠 ! )(0.50 𝑚) = 4.9 𝐽 If the floor is the reference point, 𝐸! = 𝑚𝑔ℎ = 1.0 𝑘𝑔 (9.81 𝑚/𝑠 ! )(1.50 𝑚) = 15 𝐽 The Work-­‐Energy Theorem When work is performed on an object, its speed changes and so it’s kinetic energy does too. We can say that the net work done on an object equals the change in its kinetic energy: 𝑊!"# = ∆𝐸!
Want more info? It’s a bit complicated, but here it is: https://www.youtube.com/watch?v=cwO6AkcGRBI The Law of Conservation of Energy The Law of Conservation of energy states that in a mechanical system, the total energy of the system remains constant. The energy can be transformed from one form to another, but it cannot be created or destroyed. For an example where an object is being dropped near the Earth’s surface, at any given time we can say that the value of the total energy is constant !
𝐸!"! = ! 𝑚𝑣 ! + 𝑚𝑔ℎ = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Some good examples of energy transformation and conservation: https://www.youtube.com/watch?v=Jnj8mc04r9E https://www.youtube.com/watch?v=7K4V0NvUxRg Practice: 1) What is the kinetic energy of an 18 g bird travelling at 40. km/h? 2) How much work must be done to accelerate a 2.25 kg ball at rest to 3.64 m/s? 3) How high is a 5.6 kg crate lifted if it takes 82.4 Joules to lift it? 4) A 15 kg brick is dropped from a height of 9.0 meters. Use conservation of energy to find: a) the speed of the brick when it is half way to the ground b) the speed of the brick just before it hits the ground 5) A skateboarder and their board have a combined mass of 58 kg. If the skateboarder starts up a ramp with a velocity of 6.5 m/s, how high up the ramp should they go? Homework: p. 143, p. 144, p. 147, pp. 148 -­‐149 ASIDE: We learned previously that in a closed system, momentum is always conserved. This is NOT true of kinetic energy. Kinetic energy can be “lost” through, for example, the d eformation of objects on impact (causing heat). We have 3 types of collisions Type Description Momentum Kinetic energy conserved? Conserved? Elastic Two objects bounce off of each YES YES other and are not deformed Inelastic Two objects bounce off of each YES NO other but are slightly deformed Totally Two objects stick together after YES NO inelasitic they collide