Understanding energy transformations and the work-energy theorem
Work and energy are fundamental concepts in physics that provide an alternative approach to analyzing motion. Instead of directly using Newton's laws and kinematics, we can often solve problems more easily by considering the work done on an object and the resulting changes in its energy. This approach is particularly powerful for complex systems and provides deep insights into the conservation laws that govern our universe.
Work is the energy transferred to or from an object when a force is applied to it over a distance. Mathematically, work is the dot product of the force vector and the displacement vector.
Key points about work:
A person pushes a 50 kg crate with a force of 200 N at an angle of 30° below the horizontal, moving it 5 meters across a floor. Calculate the work done by the pushing force.
\[W = Fd \cos \theta = 200 \text{ N} \times 5 \text{ m} \times \cos(30°)\]
\[W = 200 \times 5 \times 0.866 = 866 \text{ J}\]
Kinetic energy is the energy an object possesses due to its motion. It depends on both mass and velocity.
Kinetic Energy:
KE = ½mv²
Where:
Example: Calculate the kinetic energy of a 1500 kg car traveling at 20 m/s.
KE = ½mv² = ½ × 1500 kg × (20 m/s)² = 750 kg × 400 m²/s² = 300,000 J = 300 kJ
The work-energy theorem states that the net work done on an object equals the change in its kinetic energy.
Work-Energy Theorem:
Wnet = ΔKE = KEfinal - KEinitial = ½m(vf² - vi²)
Where:
Example: A 2 kg ball initially moving at 5 m/s speeds up to 8 m/s. How much work was done on the ball?
Wnet = ΔKE = ½m(vf² - vi²) = ½ × 2 kg × ((8 m/s)² - (5 m/s)²) = 1 kg × (64 m²/s² - 25 m²/s²) = 39 J
Potential energy is stored energy that an object has due to its position or condition. Two common types are gravitational potential energy and elastic potential energy.
This is the energy an object possesses due to its height above a reference level.
Gravitational Potential Energy:
PEg = mgh
Where:
Example: A 5 kg object is lifted 2 meters above the ground. Calculate its gravitational potential energy relative to the ground.
PEg = mgh = 5 kg × 9.8 m/s² × 2 m = 98 J
This is the energy stored in elastic materials when they are stretched or compressed.
Elastic Potential Energy:
PEe = ½kx²
Where:
Example: A spring with a spring constant of 400 N/m is compressed by 0.1 meters. Calculate the elastic potential energy stored in the spring.
PEe = ½kx² = ½ × 400 N/m × (0.1 m)² = 200 N/m × 0.01 m² = 2 J
The law of conservation of energy states that energy cannot be created or destroyed but can only be transferred or converted from one form to another. In a closed system, the total energy remains constant.
Conservation of Mechanical Energy:
KEi + PEi = KEf + PEf
or
Etotal = constant
Where:
In the presence of non-conservative forces like friction, the mechanical energy is not conserved, but the total energy (including heat and other forms) is still conserved.
Example: A 2 kg ball is dropped from a height of 5 meters. Neglecting air resistance, what is its speed just before it hits the ground?
Using conservation of energy:
KEi + PEi = KEf + PEf
0 + mgh = ½mv² + 0
mgh = ½mv²
gh = ½v²
v² = 2gh
v = √(2gh) = √(2 × 9.8 m/s² × 5 m) = √98 m²/s² ≈ 9.9 m/s
Power is the rate at which work is done or energy is transferred. It tells us how quickly energy is being used or produced.
Power:
P = W/t = Fv
Where:
Example: A 60 kg person climbs a staircase that is 5 meters high in 10 seconds. Calculate the power output.
Work done = mgh = 60 kg × 9.8 m/s² × 5 m = 2940 J
Power = W/t = 2940 J / 10 s = 294 W
Understanding work and energy is essential for many applications:
Coming soon! Check back for practice problems and detailed solutions.