Force, Work, and Energy REvision Notes BTER Polytechnic Classes

 

Force, Work, and Energy 

This chapter deals with the fundamental concepts of force, work, energy, and power. These are essential topics in classical mechanics and play a crucial role in understanding how physical systems evolve and how energy is transferred and conserved.

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2.1 Force, Momentum - Statement and Derivation of Conservation of Linear Momentum

Force:

• Force is any interaction that causes an object to change its state of motion or rest. It is a vector quantity and has both magnitude and direction.
• Unit: The SI unit of force is the Newton (N), where $1 \, \text{N} = 1 \, \text{kg} \cdot \text{m/s}^2$.

Momentum:

• Momentum (p) of an object is the product of its mass and velocity: $$p = m \cdot v$$ Where:
  • $p$ is momentum (kg·m/s),
  • $m$ is mass (kg),
  • $v$ is velocity (m/s).

Conservation of Linear Momentum:

• The conservation of linear momentum states that the total momentum of a closed system remains constant if no external forces act on it. This principle is a consequence of Newton's third law of motion.

• Statement: In an isolated system, the total linear momentum before any interaction is equal to the total linear momentum after the interaction.

• Mathematical Derivation:

  • Let there be two objects $m_1$ and $m_2$ with velocities $v_1$ and $v_2$ before a collision, and velocities $v_1'$ and $v_2'$ after the collision.
  • The total momentum before the collision is: $$p_{\text{initial}} = m_1 v_1 + m_2 v_2$$
  • The total momentum after the collision is: $$p_{\text{final}} = m_1 v_1' + m_2 v_2'$$
  • According to the principle of conservation of momentum: $$p_{\text{initial}} = p_{\text{final}}$$ Thus: $$m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'$$

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2.2 Applications such as Recoil of Gun, Rockets

Recoil of a Gun:

• When a gun is fired, the bullet gains momentum in one direction, and the gun experiences an equal and opposite momentum (recoil).
• Using conservation of momentum:
  • Before firing, both the gun and the bullet are at rest, so total momentum is zero.
  • After firing, the total momentum must still be zero.
  • If the mass of the bullet is $m_b$, its velocity is $v_b$, and the mass of the gun is $m_g$, the recoil velocity of the gun is $v_g$.
  • Using conservation of momentum: $$0 = m_b v_b + m_g v_g$$
  • Thus, the recoil velocity of the gun is: $$v_g = - \frac{m_b v_b}{m_g}$$

Rockets:

• Rockets move based on the principle of conservation of momentum. The rocket ejects mass (exhaust gases) in the opposite direction, and this creates a thrust that propels the rocket forward.
• The ejected gases have momentum in one direction, and the rocket has momentum in the opposite direction, conserving total momentum.

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2.3 Work Concept and Units

Work:

• Work is done when a force is applied to an object, and the object moves in the direction of the force.
• Formula: $$W = F \cdot d \cdot \cos \theta$$ Where:
  • $W$ is the work done (Joules, J),
  • $F$ is the force applied (Newtons, N),
  • $d$ is the displacement (meters, m),
  • $\theta$ is the angle between the force and the displacement vectors.

Units:

• The SI unit of work is the Joule (J), where: $$1 \, \text{J} = 1 \, \text{N} \cdot \text{m}$$

Examples of Zero Work, Positive Work, and Negative Work:

1. Zero Work: If there is no displacement (or the force is perpendicular to the displacement), no work is done. For example, if you push a wall with force but it does not move, no work is done.

2. Positive Work: When the force and displacement are in the same direction, work is positive. For example, lifting a book from the ground involves positive work since the applied force (upward) and the displacement (upward) are in the same direction.

3. Negative Work: When the force and displacement are in opposite directions, work is negative. For example, when a person is stopping a moving car, the force applied to stop it is in the opposite direction to the car's displacement, so negative work is done.

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2.4 Energy and Its Units

Energy:

• Energy is the capacity to do work. It exists in various forms, such as kinetic energy, potential energy, thermal energy, etc.
• Units: The SI unit of energy is the Joule (J), the same as work.

2.4.1 Kinetic Energy, Gravitational Potential Energy, and Mechanical Energy

1. Kinetic Energy (KE): The energy possessed by an object due to its motion.

   • Formula: $$KE = \frac{1}{2} m v^2$$ Where:
     • $m$ is the mass (kg),
     • $v$ is the velocity (m/s).

2. Gravitational Potential Energy (PE): The energy possessed by an object due to its position in a gravitational field.

   • Formula: $$PE = mgh$$ Where:
     • $m$ is the mass (kg),
     • $g$ is the acceleration due to gravity (9.8 m/s$^2$),
     • $h$ is the height above the reference point (m).

3. Mechanical Energy: The sum of kinetic and potential energy in a system.

   • Formula: $$E_{\text{mechanical}} = KE + PE$$

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2.5 Conservation of Mechanical Energy for Freely Falling Bodies

• For a freely falling object (ignoring air resistance), the total mechanical energy (kinetic + potential) remains constant.

• As the object falls:

  • At the top, the object has maximum potential energy and zero kinetic energy.
  • As it falls, potential energy decreases, and kinetic energy increases.
  • At the bottom, the object has maximum kinetic energy and zero potential energy.

• The total mechanical energy at any point is:

  $$E = KE + PE = \text{constant}$$

  Thus, the sum of the potential and kinetic energy is conserved during the fall.

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2.6 Power and Its Units

Power:

• Power is the rate at which work is done or energy is transferred. It is the amount of energy consumed or work done per unit time.
• Formula: $$P = \frac{W}{t}$$ Where:
  • $P$ is the power (Watts, W),
  • $W$ is the work done (Joules, J),
  • $t$ is the time taken (seconds, s).

Units:

• The SI unit of power is the Watt (W), where $1 \, \text{W} = 1 \, \text{J/s}$.

2.6.1 Power and Work Relationship

• The relationship between power and work is derived from the formula $P = \frac{W}{t}$. If work is done over a certain period of time, power quantifies how quickly the work is done.

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2.6.2 Calculation of Power (Numerical Problems)

1. Example: A motor lifts a 10 kg box vertically by 5 m in 10 seconds. Calculate the power exerted by the motor.

   • First, calculate the work done (W):

     $$W = F \cdot d = mgh = 10 \cdot 9.8 \cdot 5 = 490 \, \text{J}$$

   • Now, calculate the power (P):

     $$P = \frac{W}{t} = \frac{490}{10} = 49 \, \text{W}$$

Thus, the power exerted by the motor is 49 watts.

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Summary of Key Concepts:

1. Force: A vector quantity that causes an object to move or change motion.
2. Momentum: The product of an object's mass and velocity. Conserved in an isolated system.
3. Work: Force applied over a distance. Can be positive, negative, or zero.
4. Energy: The capacity to do work. Includes kinetic energy, potential energy, and mechanical energy.
5. Conservation of Mechanical Energy: Total energy (kinetic + potential) remains constant in the absence of external forces.
6. Power: The rate at which work is done or energy is transferred.

These concepts are fundamental in understanding classical mechanics and form the basis of many applications in engineering and physics.