5.3 mass and weigjt

Key Equations and Concepts

• Weight Equation: The net force in the vertical (y) direction when only gravity and normal force are acting, where w is weight and mg represents mass times gravitational acceleration. \sum F_y = N - w = N - mg

  • In free fall (no normal force), the net force is just weight: \sum F_y = -w = -mg

  • For objects at rest or constant velocity on a horizontal surface, N = w = mg

• Weight Variations by Location:

  • Moon: g_{moon} \approx \frac{1}{6} g_{earth} ; thus, an object's weight on the moon is about one-sixth of its Earth weight, while its mass remains the same.

  • Jupiter: Due to its larger mass and gravitational field, weight on Jupiter would be significantly greater than on Earth, but the object's intrinsic mass does not change.

Mass vs Weight

• Mass: A fundamental intrinsic property of an object, representing its inertia (resistance to changes in motion). It is a scalar quantity and remains constant regardless of location.

  • Standard unit: kilograms (kg).

• Weight: The force exerted on an object due to gravity. It is a vector quantity, directed downwards, and varies with the local gravitational field.

  • Standard unit: Newtons (N).

  • Typically recorded in pounds in the imperial system (1 lb = 4.45 \text{ N}).

• Conversions:

  • weight = mg

  • mass = \frac{weight}{g}

Conversion Factors

• 1 \text{ lb} = 4.45 \text{ N}

• 1 \text{ N} \approx 0.22 \text{ lbs}

• 1 \text{ kg} \approx 2.2 \text{ lbs}

• 1 \text{ lb} \approx 0.454 \text{ kg}

• 1 \text{ kg} = 1000 \text{ g}

• Standard gravitational acceleration on Earth (g): 9.8 \text{ m/s}^2

Apparent Weight (W_{app})

• Definition: The magnitude of the supporting contact forces (normal force) acting on an object. It's what a scale-reads.

• In equilibrium (at rest or constant velocity): W_{app} = W = mg (the normal force equals the actual weight).

Apparent Weight Change with Acceleration (Elevator Example)

• According to Newton's Second Law ( \sum F = ma ), the normal force (apparent weight) changes with acceleration:

  • Acceleration upwards (or slowing down while moving downwards): The supporting force must be greater than gravity to cause an upward net force. W_{app} = N = W + ma

  • Acceleration downwards (or slowing down while moving upwards): The supporting force is less than gravity, as gravity contributes to the downward acceleration. W_{app} = N = W - ma

Practical Examples

• Scale Reading: If an elevator accelerates upwards or slows down while moving downwards, the scale reads higher than the object's actual weight. Conversely, if it accelerates downwards or slows down while moving upwards, the scale reads lower.

• During free fall: If the elevator cable breaks, both the person and the scale accelerate downwards at g. There is no supporting contact force because the scale falls at the same rate as the person, resulting in W_{app} = 0 . This zero normal force causes the sensation of weightlessness.

Notable Example of Weightlessness

• Astronauts experience prolonged weightlessness in orbit because they are in a continuous state of free fall around the Earth. The gravitational force is still present, but the absence of a supporting surface makes their apparent weight zero.

Kinematic Application

• Kinematic equations can be used to analyze the motion (stopping time, speed, and acceleration changes) of moving systems like elevators, which then affect apparent weight calculations:

  • \Delta t = \frac{V_{final} - V_{initial}}{a}

  • This 'a' (acceleration from kinematics) is the same 'a' used in the apparent weight formulas to determine the change in normal force.

Weightlessness is the sensation experienced when there is no supporting contact force acting on an object, making its apparent weight zero. This often occurs during free fall, where an object accelerates downwards at the same rate as its surroundings, such as in a falling elevator or when astronauts are in orbit around Earth, continuously falling without a supporting surface.