7.2 Kinetic Energy and the Work-Energy Theorem

Kinetic energy and work are closely linked concepts in physics. The work-energy theorem shows how the work done on an object changes its kinetic energy, while energy conservation principles explain how energy transforms between different forms.

Forces play a crucial role in work-energy problems. The net force on an object determines its acceleration and change in kinetic energy. Understanding these relationships helps us analyze and predict the motion of objects in various scenarios.

Kinetic Energy and Work-Energy Theorem

Work-energy theorem calculations

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• Work-energy theorem states the net work done on an object equals the change in its kinetic energy
  • Equation: $W_{net} = \Delta KE = KE_f - KE_i$
    • $W_{net}$ represents the net work done on the object
    • $KE_f$ represents the final kinetic energy
    • $KE_i$ represents the initial kinetic energy
• Work is the product of the force applied to an object and the displacement of the object in the direction of the force
  • Equation: $W = F \cdot d \cdot \cos\theta$
    • $F$ represents the force applied
    • $d$ represents the displacement
    • $\theta$ represents the angle between the force and the displacement
    • Example: Pushing a box 5 m with a force of 20 N at a 30° angle to the horizontal
• Kinetic energy is the energy an object possesses due to its motion
  • Equation: $KE = \frac{1}{2}mv^2$
    • $m$ represents the mass of the object
    • $v$ represents the velocity of the object
    • Example: A 2 kg ball moving at 3 m/s has a kinetic energy of 9 J

Kinetic energy in energy transfer

• Energy transfer occurs when work is done on an object, changing its kinetic energy
  • Positive work increases kinetic energy (acceleration)
  • Negative work decreases kinetic energy (deceleration)
  • Example: A car accelerating from rest to 60 km/h
• Conservation of energy states that in the absence of non-conservative forces, the total energy of a system remains constant
  • Equation: $\Delta KE = -\Delta PE$
    • $\Delta PE$ represents the change in potential energy
    • Example: A roller coaster at the top of a hill has high potential energy and low kinetic energy, while at the bottom it has low potential energy and high kinetic energy
• Power is the rate at which work is done or energy is transferred
  • Equation: $P = \frac{W}{\Delta t}$
    • $P$ represents power
    • $W$ represents work
    • $\Delta t$ represents the time interval
    • Example: A 100 W light bulb converts 100 J of electrical energy into light and heat every second

Forces in work-energy problems

• Net force is the vector sum of all forces acting on an object
  • If the net force is zero, the object's velocity remains constant (Newton's first law)
  • If the net force is non-zero, the object's velocity changes (Newton's second law)
  • Example: A book resting on a table has a net force of zero, while a falling book has a net force equal to its weight
• Acceleration is the rate of change of an object's velocity
  • Equation: $a = \frac{F_{net}}{m}$
    • $a$ represents acceleration
    • $F_{net}$ represents the net force
    • Example: A 1000 kg car accelerating at 2 m/s² experiences a net force of 2000 N
• Work-kinetic energy relationship states the work done by the net force on an object equals the change in its kinetic energy
  • Equation: $W_{net} = F_{net} \cdot d \cdot \cos\theta = \Delta KE$
  • Example: A 5 kg box pushed 3 m by a 10 N force at a 60° angle to the horizontal experiences a change in kinetic energy of 15 J
• Energy conservation states that in the absence of non-conservative forces, the total energy (kinetic + potential) remains constant
  • Equation: $\Delta KE + \Delta PE = 0$
  • Example: A pendulum swinging back and forth, constantly converting kinetic energy to potential energy and vice versa
  • Friction is a non-conservative force that can cause energy to be dissipated as heat

Energy, Momentum, and Motion

• Kinetic energy and momentum are both related to an object's motion
  • Momentum is the product of an object's mass and velocity (p = mv)
  • Kinetic energy depends on the square of velocity (KE = 1/2mv²)
• The relationship between mass, velocity, and energy affects various physical phenomena
  • A change in an object's velocity results in changes to both its kinetic energy and momentum
• Energy conversion occurs when one form of energy is transformed into another
  • Example: In a hydroelectric dam, the potential energy of water is converted to kinetic energy of the turbines, which is then converted to electrical energy