7. Degrees of Freedom, Free Body Diagrams, & Fictitious Forces

The script begins with an introduction to MIT OpenCourseWare, emphasizing its mission to provide free, high-quality educational resources. It encourages donations to support this initiative and directs interested parties to the website for more materials. The professor then discusses 'muddy cards,' a feedback tool used to gather student impressions on lectures. The cards revealed that students appreciated practical demonstrations and clear explanations with examples, particularly regarding the computation of angular momentum from different reference points. The professor addresses a common misconception about the independence of vectors from coordinate systems, explaining the concept of angular momentum relative to different points and clarifying its dependency on the choice of reference point.

This paragraph delves into the concept of degrees of freedom in dynamics, defined as the number of independent coordinates needed to describe motion. The professor outlines a formula to calculate degrees of freedom, which involves the number of rigid bodies, particles, and constraints present in a system. Using a wheel as an example, the professor illustrates how to identify constraints and calculate degrees of freedom, emphasizing the importance of understanding constraints like no motion in certain directions and no slip conditions to simplify the problem and derive the correct equations of motion.

The script continues with an analysis of a wheel on a slope, considering it as a rigid body and discussing the preliminary steps in setting up a coordinate system to study its motion. The focus then shifts to identifying constraints, such as the inability to move in certain directions and assumptions about the wheel's behavior, like no slipping. The professor calculates the number of degrees of freedom and explains how to use these insights to develop a free body diagram, which includes forces like gravity, normal force, and a tangential force that might prevent slipping.

The dynamics of a hockey puck wrapped with a string is the subject of this paragraph. The professor explains how to determine the number of degrees of freedom for the system, taking into account the constraints imposed by the surface and the string. The analysis involves considering the inability to move through the surface and the restrictions on rotation about certain axes. The professor then discusses the need for multiple equations of motion to fully describe the system's behavior, highlighting the importance of selecting appropriate coordinates and understanding the constraints.

The script presents a scenario involving a rod leaning against a wall, focusing on the concepts of translation and rotation. The professor discusses the constraints at the points of contact between the rod and the wall, explaining how these constraints affect the degrees of freedom. The analysis includes identifying the center of mass, the importance of the instantaneous center of rotation (ICR), and the implications for the motion of the rod. The professor emphasizes the need for a single coordinate to describe the rod's motion and the usefulness of the ICR in simplifying the equations of motion.

In this paragraph, the professor reflects on the unknowns present in the free body diagram of the rod from the previous example. The discussion revolves around identifying the forces acting on the rod, such as normal forces and gravity, and considering the implications of a frictionless environment. The professor highlights the need to find relationships between these unknowns to solve the equations of motion, mentioning the use of the instantaneous center of rotation and the relationship between torque and angular momentum.

The professor introduces a new example involving two carts connected by a spring and a dashpot, constrained by the floor. The focus is on calculating the degrees of freedom for this system, considering the constraints and the independent coordinates needed to describe the motion of both carts. The professor suggests that there are 10 constraints and that two independent coordinates are necessary, recommending the use of coordinates starting from the static equilibrium position for simplicity.

This paragraph discusses the assignment of free body diagrams for the two-cart system with springs and dashpots. The professor explains the process of determining the direction of forces exerted by the springs and dashpots on each cart, emphasizing the importance of considering the system's positive motions to deduce these forces. The explanation includes the principles of superposition and the linear nature of the forces involved, leading to the formulation of equations of motion for the system.

The script shifts focus to the concept of fictitious forces, which are introduced as useful but potentially dangerous tools in physics. The professor explains that fictitious forces arise from considering the acceleration of a reference frame and can be treated as forces in the equations of motion. The paragraph includes a simple example of an elevator accelerating upwards, illustrating how to calculate the apparent weight of a person inside using the concept of fictitious forces.

Building on the concept of fictitious forces, the professor presents a more complex example involving a mass attached to a rotating shaft. The analysis requires understanding the acceleration components in a rotating reference frame and identifying the forces that can be treated as fictitious. The professor breaks down the acceleration into its cylindrical coordinate components and discusses how to use these components to quickly estimate the torques and forces involved in the system.

In this paragraph, the professor demonstrates how to use fictitious forces to quickly estimate torques and bending moments in a rotating system. The example involves a rotating shaft with a mass attached, and the professor shows how to calculate the torque caused by the centrifugal force and the bending moment it creates. The discussion highlights the importance of understanding the direction and magnitude of these fictitious forces and their implications for the design and analysis of rotating systems.

The script concludes with a cautionary note about the use of fictitious forces. The professor emphasizes the importance of distinguishing between real forces, such as gravity or contact forces, and fictitious forces, which are accelerations treated as forces for convenience. The professor advises that while fictitious forces can enhance intuition and provide quick estimates, one must always return to fundamental principles for rigorous analysis.

The final paragraph provides a brief preview of a mechanical shaker demonstration that will be discussed in the next session. The professor encourages students to fill out their muddy cards with questions or feedback, indicating a commitment to interactive and engaging teaching methods. The script ends with an anticipation of further exploration into the dynamics of mechanical systems.