Rotational Dynamics: Torque, Center of Gravity, and Rolling Motion Lecture 12

Recap: Planetary Gravity and Orbital Motion

Gravitational Attraction and Free-Fall Acceleration

Gravity is a fundamental force that governs the motion of objects with mass. The gravitational force between a planet and an object on its surface depends on their masses and the distance between their centers.

• Newton's Law of Universal Gravitation: The force between a planet (mass $M_{planet}$) and an object (mass $m$) at the surface (distance $R_{planet}$ from the center) is given by: $F_{planet ext{ on }m} = \frac{GM_{planet}m}{R_{planet}^2}$

• Free-fall acceleration at the surface: $g_{planet} = \frac{GM_{planet}}{R_{planet}^2}$

Orbital Motion

Satellites in circular orbits experience gravitational force as the centripetal force keeping them in orbit.

• Orbital speed: $v = \sqrt{\frac{GM}{r}}$

• Orbital period: $T^2 = \left(\frac{4\pi^2}{GM}\right) r^3$

Linear vs. Rotational (Circular) Motion

Many variables and equations for linear motion have analogs in rotational motion.

Linear Motion	Circular Motion
Position ($x$)	Angle ($\theta$)
Velocity ($v_x = \frac{\Delta x}{\Delta t}$)	Angular velocity ($\omega = \frac{\Delta \theta}{\Delta t}$)
Acceleration ($a_x = \frac{\Delta v_x}{\Delta t}$)	Angular acceleration ($\alpha = \frac{\Delta \omega}{\Delta t}$)
$\Delta x = v_x \Delta t$ (constant velocity)	$\Delta \theta = \omega \Delta t$ (constant angular velocity)
$\Delta x = v_{0x} \Delta t + \frac{1}{2} a_x (\Delta t)^2$ (constant acceleration)	$\Delta \theta = \omega_0 \Delta t + \frac{1}{2} \alpha (\Delta t)^2$ (constant angular acceleration)

Section 7.3: Torque

Definition and Calculation of Torque

Torque ($\tau$) is the rotational equivalent of force. It measures the ability of a force to cause an object to rotate about an axis (pivot point).

• Factors affecting torque:

  • Magnitude of the force ($F$)

  • Distance from the pivot ($r$)

  • Angle ($\phi$) between the force and the radial line from the pivot

• Mathematical definition: $\tau = r F_{\perp}$ or $\tau = r F \sin \phi$

• Units: Newton-meter (N·m)

• Radial line: The line from the pivot to the point where the force is applied.

• Angle $\phi$: Measured from the radial line to the direction of the force.

Moment Arm (Lever Arm) and Line of Action

• Moment arm: The perpendicular distance from the pivot to the line of action of the force.

• Line of action: The line along which the force acts.

• Alternative torque formula using moment arm $l$: $\tau = l F$

Section 7.4: Gravitational Torque and Center of Gravity

Gravitational Torque

Gravity acts on every particle of an object, but the net gravitational force can be considered to act at a single point called the center of gravity.

• Gravitational torque: $\tau = r_{cg} W \sin \theta$ where $r_{cg}$ is the distance from the pivot to the center of gravity, $W$ is the weight, and $\theta$ is the angle.

• For extended objects, the torque due to gravity can be calculated as if all the weight acts at the center of gravity.

Center of Gravity

• The point where the total gravitational torque is zero when used as the pivot.

• For a system of particles, the center of gravity's position $x_{cg}$ is: $x_{cg} = \frac{\sum m_i x_i}{\sum m_i}$

• Similarly for the $y$-coordinate: $y_{cg} = \frac{\sum m_i y_i}{\sum m_i}$

Example: Balancing a Seesaw

• To balance, the combined center of gravity of the two children must be at the pivot.

• If Noah (22 kg) sits 1.6 m from the pivot, Emma (25 kg) must sit at $x$ such that: $0 = \frac{22 \times (-1.6) + 25 \times x}{22 + 25}$ Solving gives $x = 1.4$ m.

Section 7.5: Rotational Dynamics and Moment of Inertia

Newton's Second Law for Rotation

Torque causes angular acceleration, analogous to how force causes linear acceleration.

• Newton's second law for rotation: $\tau_{net} = I \alpha$ where $I$ is the moment of inertia and $\alpha$ is angular acceleration.

• Moment of inertia ($I$): The rotational equivalent of mass, depends on mass distribution relative to the axis. $I = \sum m_i r_i^2$

• Units: kg·m2

Moments of Inertia for Common Shapes

Object	Axis	Moment of Inertia ($I$)
Rod (length $L$)	Center	$\frac{1}{12}ML^2$
Rod (length $L$)	End	$\frac{1}{3}ML^2$
Solid cylinder or disk (radius $R$)	Center	$\frac{1}{2}MR^2$
Thin spherical shell (radius $R$)	Diameter	$\frac{2}{3}MR^2$

Physical Interpretation

• Objects with mass farther from the axis have higher moments of inertia and are harder to spin.

• Example: A tightrope walker uses a long pole to increase moment of inertia, reducing angular acceleration and improving stability.

Section 7.6: Using Newton's Second Law for Rotation

Problem-Solving Approach

• Model the object and identify the axis of rotation.

• Identify forces and their distances from the axis.

• Calculate torques and determine their signs.

• Apply $\tau_{net} = I \alpha$ to solve for unknowns.

• Use rotational kinematics as needed.

Constraints with Ropes and Pulleys

• If a rope does not slip on a pulley, the linear acceleration of the rope matches the tangential acceleration of the pulley's rim.

• Constraint equations: $v_{y} = -\omega R$ $a_{y} = -\alpha R$

Section 7.7: Rolling Motion

Rolling Without Slipping

Rolling motion combines rotation about an axis and translation of the center of mass.

• For one revolution, the center moves forward by the circumference: $\Delta x = 2\pi R$

• Rolling constraint: $v = \omega R$

• The point at the bottom of a rolling object is instantaneously at rest relative to the surface.

Summary of Key Concepts

• Torque: $\tau = rF\sin\phi$

• Moment of inertia: $I = \sum m_i r_i^2$

• Newton's second law for rotation: $\tau_{net} = I\alpha$

• Center of gravity: $x_{cg} = \frac{\sum m_i x_i}{\sum m_i}$

• Rolling constraint: $v = \omega R$

Additional info: These notes expand on the provided slides and text, filling in standard definitions, formulas, and context for clarity and completeness.