PHYS101: Introduction to Mechanics

5a. Compare and contrast the physical properties associated with linear motion and rotational motion

• Define the rotational angle, arc length, and radius of curvature.
• What is the meaning of the unit radians (rad)?

In rotational motion we deal with two dimensional motion. Unlike with linear motion, we need to define angles and distances associated with circular motion. To understand circular or rotational motion, picture a spinning disk, such as the picture of a CD in Figure 6.2. This figure shows a CD with a line drawn from the center to the edge. All the points along this line travel the same travel the same angle in the same amount of time as the CD spins. This is called the rotational angle, which is defined as:

$\Delta \theta=\frac{\Delta s}{r}$

where $\Delta s$ is the circumference traveled and $r$ is the radius.

We call the curvature traveled ($\Delta s$) the arc length, and we call the radius ($r$) the radius of curvature. 

When describing angles, we often use the unit radian, abbreviated as rad or r. We define radians as:

1 revolution = $2\pi$ rad

Radians can be converted to the more familiar degrees. Review Table 6.1: Comparison of Angular Units for conversions between radians and degrees. 

5b. Explain why an object moving at a constant speed in a circle is accelerating

• Define angular velocity. How is angular velocity related to linear velocity?
• Define centripetal acceleration. In what direction is centripetal acceleration?

We define angular velocity, $\omega$ (the Greek letter omega), as the rate at which the angle changes while an object is rotating. We can write it as:

$\omega=\frac{\Delta \theta}{\Delta t}$

where $\Delta \theta$ is the change in angle and $\Delta t$ is the change in time.

We can relate angular velocity to linear velocity using the following relation:

$v = r \omega$

with $r$ being the radius of curvature.

Review the derivation of how angular velocity relates to linear velocity in equations 6.6, 6.7, 6.8, and 6.9 in Rotational Angle and Angular Velocity.

When an object moves in a circular motion, the net change in velocity points toward the center of the object, based on the vector addition of the velocity vectors as the object rotates. Whenever there is a change in velocity, there is acceleration in the direction of the change of the change in velocity. This means that for a rotating object, there is an acceleration toward the center of the object. We call this centripetal acceleration, $a_c$.

The equation for centripetal acceleration is:

$a_c=\frac{v^2}{r}$

where $v$ is speed and $r$ is radius.

Review an example of this in Figure 6.8. In this example, a disk is rotating at a constant speed. As the disk rotates, the velocity vector at any given point on the disk changes because the direction changes. As shown in the free body diagram at the top of the figure, the velocity vectors add to make a net velocity vector toward the center of the disk. This leads to centripetal acceleration because there is a net change in acceleration toward the disk.

5c. Apply Newton's Law of Gravity

• State Newton's Universal Law of Gravity.
• How does gravitational force relate to object mass? Distance between objects?
• Calculate the acceleration of an object due to gravity.

Newton's Universal Law of Gravity states that all objects in the universe attract each other in straight force lines between them. Review an example of how two objects exert gravitational forces on each other in a straight line in Figure 6.21. 

The force between two objects is directly related to the product of the masses. It is inversely proportional to the distance between the objects squared.

For two objects with masses $M$ and $m$ and radius $r$, this can be written as:

$F = G\frac{Mm}{r^2}$

where $G$ is the gravitational constant, $6.674\times 10^{-11} \mathrm{\frac{Nm^2}{kg^2}}$

Review a worked example of using Newton's Universal Law of Gravity to calculate the acceleration of an object due to gravity in Example 6.6: Earth's Gravitational Force is the Centripetal Force Making the Moon Move in a Curved Path.

5d. Solve problems involving planets and satellites

• State Kepler's Three Laws.
• Use Kepler's Laws to determine how long it takes for a satellite to make its orbit.

Kepler's Laws of Planetary Motion describe the motion of planets around the sun. They can also be extended to describe the motion of satellites around planets.

Kepler's First Law of Planetary Motion states that planets move around the sun in an ellipse shaped orbit with the sun at the center of the ellipse. Review Figure 6.29.

Kepler's Second Law of Planetary Motion states that planets move such that a point on the planet sweeps an equal area in equal times. Review Figure 6.30.

Kepler's Third Law of Planetary Motion relates the time it takes for two planets to revolve around the sun to their distances from the sun in the following equation:

$\frac{T_1^2}{T_2^2}=\frac{r_1^3}{r_2^3}$

where $T_1$ and $T_2$ are periods and $r_1$ and $r_2$ are radii for planets 1 and 2.

We can use Kepler's Third Law to solve problems to determine the period for planetary or satellite orbits.

Review a worked example of using the equation from Kepler's Third Law to determine the period of a satellite in Example 6.7: Find the Time for One Orbit of an Earth Satellite.

5e. Explain what it means when an astronaut in earth's orbit is described as being "weightless"

• Define microgravity.
• Is an astronaut in space really experiencing zero gravity? Why or why not?

Microgravity occurs when the net acceleration on an object is significantly less than the net acceleration the object would experience on the earth's surface. It occurs for astronauts in orbit. The astronauts are in free fall toward the earth. This means they are experiencing the earth's gravitational force and are therefore not "weightless" or at "zero gravity". 

Unit 5 Vocabulary

• Acceleration
• Angular velocity
• Arc length
• Centripetal acceleration
• Circular motion
• Gravitational force
• Kepler's First Law of Planetary Motion
• Kepler's Second Law of Planetary Motion
• Kepler's Third Law of Planetary Motion
• Linear motion
• Microgravity
• Newton's Universal Law of Gravity
• Object mass
• Radian
• Radius of curvature
• Rotational angle
• Zero gravity