KNES 361 L14

Lecture 14: Angular Kinetics - Newton’s Laws of Motion

Overview

• This lecture serves as a wrap-up for the section on angular kinetics.

• Key concepts previously covered: torque, moment of inertia, angular momentum, law of conservation of angular momentum.

• The focus of the lecture is to translate Newton's three laws of motion from linear to rotational forms.

Newton's First Law of Motion

• Linear Version: An object remains at rest or moves at constant velocity unless acted upon by an unbalanced force.

• Rotational Version: A rotating body will maintain a state of constant rotational motion unless acted on by an external torque.

  • This implies that if no external torque is applied, the state of angular motion remains unchanged.

• Connection to Conservation of Angular Momentum:

  • If external torque is zero, angular motion remains constant, and thus angular momentum is conserved.

  • Example: A gymnast or skater can speed up by pulling their arms in without creating new angular momentum; they simply redistribute their body to change moment of inertia, which consequently alters angular velocity while keeping total angular momentum constant.

Newton's Second Law of Motion

• Linear Version: States the relationship between force and acceleration:

  • An unbalanced force causes acceleration that is proportional to the force, in the direction of the force, and inversely proportional to the mass of the object.

  • Formula: $a = \frac{F}{m} \text{ or equivalently } F = ma$

• Rotational Version: The relationship is expressed in terms of torque and moment of inertia:

  • Torque causes angular acceleration, and moment of inertia resists angular acceleration.

  • Formula: $\tau = I\alpha \text{ or equivalently } \alpha = \frac{\tau}{I}$

  • Interpretation: A larger torque results in greater angular acceleration, while a larger moment of inertia leads to smaller angular acceleration for the same torque.

Torque and Moment of Inertia

• Torque is defined as: $\tau = F \cdot d$

  • Where d is the perpendicular distance from the axis of rotation to the line of action of the force.

• Moment of inertia is calculated as: $I = m \cdot k^2$

  • Where k is the radius of gyration.

• Rewriting Newton's second law for angular motion with these expressions:

  • $F \cdot d = (m \cdot k^2) \cdot \alpha$

Problem Solving Using Newton's Second Law for Rotational Motion

• Example Problem: Calculating the torque in a football kick.

  • Given Data:

  • Angular acceleration: $\alpha = 453 \text{ rad/s}^2$

  • Moment of inertia (lower leg about knee): $I = 0.35 \text{ kg} \cdot m^2$

  • To find torque (muscle moment):

  • Apply the formula for Newton’s second law for rotational motion:

  • $\tau = I \cdot \alpha$

  • Substituting values: $\tau = 0.35 \cdot 453$

  • Calculation: $\tau = 158.55 \text{ N m}$

  • Result: The torque required is approximately $158.55 \text{ N m}$, representing the rotational push exerted by the muscles around the knee to accelerate the lower leg.

Newton's Third Law of Motion

• Linear Version: States that for every action, there is an equal and opposite reaction.

• Rotational Version: For every torque a body exerts on another, the second body exerts an equal and opposite torque on the first body.

  • Simplified: If you twist an object, it will exert a counteracting torque back on you, which you can feel as a reaction force.

Conclusion

• The discussion covered the angular equivalents of Newton's laws, emphasizing their applications in rotational motion and the relationships involving torque, moment of inertia, and angular acceleration.