Kinetic Energy and Work Review

Chapter 6: Kinetic Energy and Work

7.2 What is Energy?

Energy is defined as a scalar quantity associated with the state (or condition) of one or more objects. This concept incorporates several key characteristics:

  1. Transformation: Energy can be transformed from one type to another—for example, mechanical to chemical or electromagnetic to heat.

  2. Transfer: Energy can be transferred from one object to another.

  3. Conservation: The total amount of energy remains constant; energy is conserved.

7.3 Kinetic Energy

Kinetic energy, represented by the symbol K, is the energy associated with the state of motion of an object. The following points detail its key properties:

  • The expression for kinetic energy is given as: K = \frac{1}{2}mv^2 where:

    • m = mass of the object (in kg)

    • v = speed of the object (in m/s)

  • Kinetic energy increases as the speed of the object increases; specifically, the faster the object moves, the greater its kinetic energy becomes.

  • The SI unit for kinetic energy (and all types of energy) is the joule (J), defined as:
    1 \text{ joule} = 1 J = 1 \text{ kg} \times \frac{m^2}{s^2}

  • An important note is that kinetic energy is always a positive value.

Sample Problem: Locomotive Collision

In 1896 in Waco, Texas, William Crush staged a collision of two locomotives. The scenario involved:

  • Two locomotives parked at opposite ends of a 6.4-km-long track.

  • Each locomotive weighs 1.2 × 10^6 N with a constant acceleration of 0.26 m/s².

  • After calculations involving kinematics, we find:

    • The speed of each locomotive just before collision:
      v = \sqrt{v0^2 + 2a(x - x0)} = \sqrt{0 + 2(0.26 m/s^2)(3.2 \times 10^3 m)} = 40.8 m/s
      (approximately 150 km/h).

    • Mass of each locomotive requires converting weight to mass:
      m = \frac{1.2 \times 10^6 N}{9.8 m/s^2} \approx 1.22 \times 10^5 kg

    • The total kinetic energy of both locomotives just before the collision:
      K_{total} = 2 m v^2 = 2(1.22 \times 10^5 kg)(40.8 m/s)^2 \approx 2.0 \times 10^8 J \approx 200 MJ

    • This amount of kinetic energy is comparable to the energy from an exploding bomb.

7.4 Work

Work, denoted W, is defined as the energy transferred to or from an object by means of a force acting on it. The characteristics of work include:

  • Work can be positive or negative:

    • Positive Work: When energy is transferred to the object.

    • Negative Work: When energy is transferred from the object.

7.5 Work and Kinetic Energy

To calculate the work done by a force F on an object that moves through a displacement d, it is important to consider only the component of the force along the direction of the displacement. The following must be noted:

  1. A component of force that is perpendicular to the displacement performs zero work.

  2. When many forces act on an object, the net work done is the summation of the works done by all individual forces.

Work-Kinetic Energy Theorem

The theorem states that the change in kinetic energy of a particle equals the net work done on it. Specifically:

  • If the net work is positive, the kinetic energy of the particle increases by that amount, and conversely, if negative, the kinetic energy decreases.

Sample Problem: Industrial Spies and Work

In a scenario where industrial spies are moving a safe with mass m = 225 kg over a displacement d = 8.50 m:

  • The forces applied are:

    • Spy 001 pushes with a force F₁ = 12.0 N at 30 degrees downward.

    • Spy 002 pulls with a force F₂ = 10.0 N at 40 degrees upward.

  1. Calculate the net work done on the safe by forces F₁ and F₂:

    • Work done by F₁:
      W1 = F1 d \cos(30^{\circ}) = (12.0 N)(8.50 m)(\cos 30^{\circ}) \approx 88.33 J

    • Work done by F₂:
      W2 = F2 d \cos(40^{\circ}) = (10.0 N)(8.50 m)(\cos 40^{\circ}) \approx 65.11 J

    • Total Net Work:
      W = W1 + W2 = 88.33 J + 65.11 J \approx 153.44 J

  2. During this displacement, gravity and normal force do zero work:

    • W_{gravity} = mgd \cos 90^{\circ} = 0

    • W_{normal} = FN d \cos 90^{\circ} = 0

  3. Find the final speed of the safe (v) after displacement:

    • By combining equations:
      W = K - K_0 = mv - 0

    • v = \sqrt{\frac{2W}{m}} = \sqrt{\frac{2(153.44 J)}{225 kg}} \approx 1.17 m/s

7.6 Work Done by Constant Force

  • The work done by gravitational force varies based on displacement direction:

    1. When an object rises, the angle between displacement and gravitational force is 180°, and work done results in negative energy (slowing the object).

    2. When an object falls, the angle becomes 0°, leading to positive work (accelerating the object).

7.7 Work Done by Variable Force: Hooke’s Law

  • This principle states that the spring force, a variable force, is proportional to the displacement from its equilibrium position: F_s = -kx where:

    • k represents the spring constant (characterizes stiffness),

    • x is the displacement from the equilibrium position.

  • The work done by a spring depends on whether the object moves closer or farther to the relaxed position:

    1. Positive work occurs if it moves closer.

    2. Negative work occurs if it moves farther away.

Sample Problem involving Spring Force

  • A canister of mass m = 0.40 kg with an initial speed of v = 0.50 m/s compresses a spring with a spring constant of k = 750 N/m. The compression distance d can be calculated using the work-energy principle:
    K1 - K0 = -\frac{1}{2}kd^2

  • Solving the equation reveals:
    d = \sqrt{\frac{m v^2}{k}} = \sqrt{\frac{0.40 kg (0.50 m/s)^2}{750 N/m}} \approx 1.2 cm

7.9 Power

Power is defined as the time rate at which work is done by a force. Key points include:

  • If a force does an amount of work W in a time interval ∆t, the average power can be expressed as:
    P_{avg} = \frac{W}{∆t}

  • Instantaneous power (P) can be described mathematically as: P = \frac{dW}{dt} = F \cdot v where F is the force applied, and v is the velocity of the object.

    • SI Unit of Power: Watt (W), equivalent to joules per second.

Conclusion

In summary, this chapter has explored the fundamental concepts of energy and work, with key focuses on kinetic energy, the work-energy theorem, and the definition of power. Each section contains mathematical formulations and applications to illustrate the practical implications of these physical phenomena. This set of definitions and principles forms a foundational understanding of the physical interactions involving force, energy, and motion.