AP Physics 1: Circular Motion and Gravitation

Student Handout

Key Concepts

1. Uniform Circular Motion

• Definition: Motion of an object traveling in a circular path at a constant speed.

• Key Characteristics:

  • The velocity is always tangent to the circle.
  • The acceleration points toward the center of the circle (centripetal acceleration).
  • The speed remains constant, but the direction of velocity changes, causing acceleration.

• Key Equations:

  • Centripetal acceleration: $a_c=\frac{v^2}{r}$
    • $v$: Linear speed of the object.
    • $r$: Radius of the circular path.
  • Centripetal force: $F_c=\frac{m{v}^2}{r}$
    • $m$: Mass of the object.
    • $F_c$: Net force directed toward the center of the circle.

• Angular Velocity:

  • $v=r\omega$, where $\omega$ is the angular velocity in radians per second.

2. Forces in Circular Motion

• Centripetal Force: The net force required to keep an object moving in a circular path.

  • It is not a new type of force but rather the result of forces like tension, gravity, friction, or the normal force.
  • Example: For a car on a curve, friction provides the centripetal force.

• Free-Body Diagrams:

  • Always draw a free-body diagram to identify the forces acting on an object in circular motion.
  • Example: For an object on a string moving in a horizontal circle, the tension in the string provides the centripetal force.

• Vertical Circular Motion:

  • At the top of the motion: $F_{\text{net}}=T+mg=\frac{m{v}^2}{r}$.
  • At the bottom of the motion: $F_{\text{net}}=T-mg=\frac{m{v}^2}{r}$.

3. Newton's Law of Universal Gravitation

• Definition: Every mass in the universe attracts every other mass with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.

• Formula: $F_g=G\frac{m_1{m}_2}{r^2}$

  • $F_g$: Gravitational force between two objects.
  • $G$: Gravitational constant ($6.67\times {10}^{-11}{\text{N·m}}^2/{\text{kg}}^2$).
  • $m_1,m_2$: Masses of the two objects.
  • $r$: Distance between the centers of the two masses.

• Gravitational Field Strength:

  • $g=\frac{GM}{r^2}$, where $M$ is the mass of the planet.
  • $g$ is the acceleration due to gravity at a distance $r$ from the planet's center.

4. Orbital Motion

• Orbital Velocity:

  • $v=\sqrt{\frac{GM}{r}}$, where $M$ is the mass of the central object (e.g., Earth).
  • This is the speed required for an object to stay in a stable orbit.

• Orbital Period:

  • $T=2\pi \sqrt{\frac{r^3}{GM}}$, where $T$ is the time it takes for one complete orbit.

• Kepler's Laws (Qualitative Understanding):

  1. Planets move in elliptical orbits with the Sun at one focus.
  2. A line joining a planet and the Sun sweeps out equal areas in equal times.
  3. The square of a planet's orbital period is proportional to the cube of its average distance from the Sun.

Step-by-Step Problem Solving

1. Centripetal Force Problems

Example: A 2.0 kg object moves in a circle of radius 0.5 m at a speed of 3 m/s. Find the centripetal force.

• Step 1: Write the formula: $F_c=\frac{m{v}^2}{r}$.
• Step 2: Substitute values: $F_c=\frac{(2.0)(3{)}^2}{0.5}$.
• Step 3: Solve: $F_c=36\text{N}$.

2. Gravitational Force Problems

Example: Two 10 kg masses are 4 m apart. Find the gravitational force between them.

• Step 1: Write the formula: $F_g=G\frac{m_1{m}_2}{r^2}$.
• Step 2: Substitute values: $F_g=(6.67\times {10}^{-11})\frac{(10)(10)}{4^2}$.
• Step 3: Solve: $F_g=4.17\times {10}^{-10}\text{N}$.

3. Orbital Motion Problems

Example: A satellite orbits Earth at a distance of $8.0\times {10}^6$ m from the center. Find its orbital velocity.

• Step 1: Write the formula: $v=\sqrt{\frac{GM}{r}}$.
• Step 2: Substitute values:
  $v=\sqrt{\frac{(6.67\times {10}^{-11})(5.97\times {10}^{24})}{8.0\times {10}^6}}$.
• Step 3: Solve: $v=7.0\text{km/s}$.

Key Diagrams

1. Centripetal Force in Circular Motion

2. Gravitational Force Between Two Masses

3. Orbital Motion