Honors Physics: Gravitation, Work, Power, Kepler’s Laws, and Centripetal Motion Study Guide

Universal Gravitation

• Gravitational Force Definition: Gravity is defined as an attractive force that exists between any two masses.

• Universal Gravitation Equation: The magnitude of the gravitational force is given by the formula:
      $F_g = \frac{G \times M_1 \times M_2}{r^2}$

• Gravitational Constant ($G$): The value of the universal gravitational constant is $6.67 \times 10^{-11}\,\text{m}^3/\text{kg} \cdot \text{s}^2$.

• Weight: Near the surface of a planet, weight is the gravitational force acting on an object, calculated using the equation:
      $F_g = m \times g$

• Determining the Correct Radius ($r$):
  

  • Between Two Masses: The radius refers to the distance measured between the centers of the two objects.

  • On a Planet’s Surface: The radius used is the radius of the planet itself.

  • Above a Planet: The radius is calculated as the sum of the planet’s radius and the altitude or height ($r = R_{planet} + h_{eight}$).

• Escape Velocity: This is the minimum speed required for an object to leave a planet's gravitational pull without further propulsion.
  

  • Equation: $v_e = \sqrt{\frac{2 \times G \times M}{r}}$

  • Earth's Escape Velocity: Approximately $25,000\,\text{mph}$.

• Schwarzschild Radius: This is the specific radius at which the escape velocity from a mass equals the speed of light ($c$).
  

  • Equation: $r_s = \frac{2 \times G \times M}{c^2}$

  • Implication: If an object collapses to a size within its Schwarzschild radius, it results in the formation of a black hole.

Work and Power

• Work Definition: Work represents the energy transferred by a force acting on an object.

• General Work Equation: $W = F_{\parallel} \times d = F \times d \times \cos(\theta)$

• Units: Work is measured in Joules ($J$).

• Conditions for Zero Work: No work is performed if the object does not move or if the force applied is perpendicular to the displacement ($\theta = 90^{\circ}$).

• Work Against Gravity: When moving an object vertically, the path taken does not matter; only the change in height is relevant.
  

  • Equation: $W = m \times g \times h$

• Work Against Friction: To calculate work done against kinetic friction:
  

  • Equation: $W = f_k \times d = \mu \times F_N \times d$

  • Note: Use a Free Body Diagram (FBD) to determine the normal force ($F_N$).

• Work performed by a Net Force:
  

  • Equation: $W = m \times a \times d$

• Positive vs. Negative Work:
  

  • Positive Work: Occurs when the force and the displacement are in the same direction.

  • Negative Work: Occurs when the force is applied in the opposite direction of the displacement.

  • Perpendicular Force: Results in zero work done ($W = 0$).

• Power Definition: Power is the rate at which work is performed.

• Power Equations:
  

  • $P = \frac{W}{t}$

  • $P = F \times v$

• Units: Power is measured in Watts ($W$).
  

  • Conversion: $1\,\text{horsepower (hp)} = 746\,W$.

Kepler’s Laws of Planetary Motion

• Kepler’s First Law (Elliptical Orbits): Planets move in elliptical orbits with the Sun positioned at one of the two foci.
  

  • Eccentricity ($e$): This measures how elongated or "stretched" an ellipse is.

  • Value Scale: A value of $0$ represents a perfect circle, while a value approaching $1$ represents a highly elongated ellipse.

  • Earth's Eccentricity: $e = 0.0167$.

• Kepler’s Second Law (Equal Areas in Equal Times): A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This implies that planets move faster when they are closer to the Sun in their orbit.

• Kepler’s Third Law (Orbital Period Relationship): There is a constant relationship between the cube of the orbital radius and the square of the orbital period.
  

  • Equation: $\frac{r^3}{T^2} = \text{constant}$

  • Comparison Form: $\frac{T_A^2}{T_B^2} = \frac{r_A^3}{r_B^3}$

  • Key Dependencies: The mass of the planet does NOT affect the orbital period, but the mass of the Sun (or central body) DOES affect the period.

• Astronomical Unit (AU):
  

  • $1\,\text{AU}$ is defined as the average distance between the Earth and the Sun.

  • Neptune's Distance: Approximately $30.1\,\text{AU}$.

Centripetal Motion

• Circular Motion Basics: Even at a constant speed, an object moving in a circle is undergoing acceleration because its direction is constantly changing. This acceleration always points toward the center of the circle.

• Tangential Velocity: This describes the linear speed of an object along its circular path.
  

  • Equation: $v = \frac{2 \times \pi \times r}{T}$

• Centripetal Acceleration ($a_c$):
  

  • Equations:
    

    • $a_c = \frac{v^2}{r}$

    • $a_c = \frac{4 \times \pi^2 \times r}{T^2}$

• Centripetal Force ($F_c$): This is the net force causing the centripetal acceleration.
  

  • Equation: $F_c = m \times a_c = \frac{m \times v^2}{r}$

  • Physical Causes: Centripetal force can be provided by tension, gravity, friction, or the normal force.

• Vertical Circles:
  

  • Calculations involve Free Body Diagrams (FBD) and Vector Addition Theorems (VAT).

  • Direction: The centripetal acceleration ($a_c$) is always directed toward the center of the circle.

  • At the Top: Gravity acts in the same direction as the centripetal acceleration, thus "helping."

  • At the Bottom: Gravity acts in the opposite direction of the centripetal acceleration, thus "opposing" the net inward force.

• Centrifugal Force: This is considered a "non-real" or fictitious force. It is actually a perception of being pushed outward caused by an object's inertia combined with an inward push from a surface (like a car seat) or a restraint (like a rope).

• Banked Curves: On a banked road, the horizontal ($x$) component of the normal force provides the necessary centripetal force. When solving these problems, do NOT tilt the coordinate axes.