Section 4.1: Work & Mechanical Energy
Work is done when a force acts on an object to move it. Mechanical energy is the sum of an object's kinetic and potential energies.
Work: \( W = F \cdot d \cdot \cos \theta \), where \(F\) is force, \(d\) is displacement, and \(\theta\) is the angle between them.
Kinetic Energy: \( KE = \frac{1}{2} m v^2 \)
Potential Energy: \( PE = m g h \)
Mechanical Energy: \( ME = KE + PE \)
Potential Energy: \( PE = m g h \)
Mechanical Energy: \( ME = KE + PE \)
Example 1
A 5 kg box is pushed 10 m with a force of 20 N at 30° to horizontal. Calculate work done.
\( W = F d \cos \theta = 20 \cdot 10 \cdot \cos 30^\circ \approx 173.2 \text{ J} \)
Example 2
A 2 kg object is dropped from 5 m. Find its kinetic energy just before hitting the ground.
\( PE = m g h = 2 \cdot 9.8 \cdot 5 = 98 \text{ J} \)
Just before impact, \( KE = 98 \text{ J} \) (assuming no friction)
Practice Problems
- A 10 kg object is lifted 2 m vertically. Find work done by gravity and by applied force.
- A 5 kg mass moves with 3 m/s. Find its kinetic energy.
- Compute mechanical energy of a 2 kg object at height 4 m moving at 5 m/s.
- A block is pushed along a frictionless surface with 15 N force over 6 m at 0° angle. Find work done.
- Calculate the height from which a 3 kg ball should fall to reach 45 J of kinetic energy just before impact.