Newton's Laws, Forces, and Torque in Biomechanics — Comprehensive Notes
Newton's laws and basic biomechanics concepts
Plan and scope (as introduced): cover Newton's laws of motion, inertia, momentum, force, ground reaction forces, friction, drag, torque, and how these ideas apply in biomechanics and rehab settings. Real-world examples given include high jump mechanics, lifting/bench press, sprinting, and gait analysis.
Key goals:
Understand how forces cause changes in motion (or resist changes in motion).
Recognize the difference between static and dynamic situations (rest vs. motion) and how unbalanced forces drive acceleration.
Connect basic physics to everyday athletic and rehabilitation scenarios.
Newton's laws of motion
Law 1 (inertia): An object at rest stays at rest, and an object in motion stays in motion with a constant velocity unless acted upon by an external unbalanced force.
In real life, this is rarely vacuum-only: gravity, friction, and air resistance are always present and can unbalance motion.
Example from transcript: a ball rolling on carpet slows down due to friction and gravity; does not remain in motion perpetually because unbalanced forces (friction and air resistance) act on it.
Law 2 (F = m a): The acceleration of an object is proportional to the net external force acting on it and inversely proportional to its mass:
F = m a
The greater the net force on an object, the greater the acceleration (for a given mass).
Mass is typically stable for humans, so acceleration changes mainly come from changes in applied force.
Practical takeaway: to accelerate faster (e.g., from the starting blocks), you must apply more force to achieve greater acceleration.
Weight and gravity on Earth:
Gravitational acceleration is approximately g \,\approx\ 9.81\ \text{m/s}^2.
Weight is W = m g. For a 1 kg mass, this is about W \approx 9.81\ \text{N}. For a 70 kg person, W \approx 70 \times 9.81 \approx 686.7\ \text{N} \approx 700\ \text{N}.
These values show the vertical force gravity contributes to weight and the baseline force that must be overcome to change vertical motion.
Inertia, momentum, and mass moments
Inertia: resistance to changes in motion (rest or in motion).
High inertia means more force is required to change velocity.
Momentum: inertia in motion; a measure of how hard it is to stop or alter the motion once moving.
Defined as \mathbf{p} = m \mathbf{v} for linear momentum.
Momentum is zero when the object is at rest.
The greater the velocity and mass, the greater the momentum.
Mass moment of inertia (rotational inertia): describes how mass is distributed relative to an axis of rotation.
It depends on both the amount of mass and how far mass is distributed from the axis (the moment arm).
For a rotating segment, angular momentum is often described by \mathbf{L} = I \boldsymbol{\omega} where $I$ is the mass moment of inertia and $\boldsymbol{\omega}$ is angular velocity.
Linear vs rotational concepts:
Linear momentum is easier to think about because it does not involve a distance from an axis.
In rotation, both the magnitude of the force and the distance from the axis (the moment arm) determine the rotational effect.
Example analogy: a limb swinging with mass concentrated farther from the shoulder has a larger moment of inertia than if the mass were closer to the axis.
Force and net force (conflict with common definitions)
Force (broad view): an interaction that can change the motion of an object.
A simple view: a push or a pull.
There are many specific force types in biomechanics: gravitational, frictional, drag, spring, tension, torque, applied forces, etc.
Work concept (related to force): the energy transfer when a force causes displacement:
W = F \cdot d \cos\theta where $\theta$ is the angle between the force and the displacement direction.
Force vs work nuance (transcribed debate): some define force as an interaction that changes motion; others emphasize that force alone does work only when there is displacement in the force direction.
Units: Force is measured in newtons (N); work in joules (J); power in watts (W).
Static equilibrium intuition:
To hold an object still, the net force must be zero (e.g., upward force balancing downward weight). If a 70 kg person stands vertically, roughly 700 N of upward reaction force balances gravity.
If you want to move an object vertically, you must apply a net upward force greater than the weight to generate upward acceleration.
Practical link: in rehabilitation or performance settings, adjusting force magnitude (via resistance) changes acceleration and control of movement.
Forces acting on objects and their classifications
Gravitational force: downward pull on mass due to Earth’s gravity.
Normal force (ground reaction in vertical direction): contact force exerted by a surface to support the weight.
Frictional force: opposes motion between surfaces; affected by surface texture, contact area, and mass; exists in both static and kinetic forms.
Air resistance (drag): friction due to movement through a fluid (air); becomes significant at higher speeds.
Spring force: restoring force from elastic elements (muscle-tendon units, ligaments, artificial springs in devices).
Tension force: pulling force transmitted through a string, rope, or tendon; a pulling force.
Applied force: intentionally generated force by muscles or external agents.
Torque and rotational forces (see below): forces applied at a distance from an axis of rotation causing rotation.
Friction and drag interplay:
Friction depends on surface properties and mass; drag depends on velocity and fluid properties.
The interplay of gravity, friction, and drag determines the net acceleration and motion trajectory.
Gravitational vs contact forces in gait and sports:
Ground reaction force is the net force provided by the ground in response to a push off the ground; it often exceeds body weight during high-impact activities like sprinting or jumping.
Ground reaction forces (GRF) and real-world measurement
Ground reaction force (GRF): the reaction from the ground when you push against it; equals and opposite to the force you apply to the ground (third-law pair).
GRF is the net normal plus shear forces exerted by the ground on the body.
Magnitude can exceed body weight, particularly during impact or deceleration phases (e.g., landing, sprinting).
Why the ground doesn’t move much: the Earth is extremely massive; it provides a counter-force but the resultant acceleration of the Earth is negligible. The interaction is about force exchange, not movement of the Earth.
Force platforms: tools used to measure GRF; in gait analysis and sprinting, GRF data helps assess loading patterns, injury risk, and rehabilitation progress.
Practical intuition:
To move upward from rest, the ground must push back with a force greater than weight (gravity) to initiate upward acceleration.
In midair or landing, the required GRF to decelerate vertical velocity is larger than static weight because you are changing velocity (kinetic energy must be dissipated).
Standing in a chair: there is a small ground reaction force through the feet; the chair provides an opposing reaction through the seating surface.
Force, motion, and practical biomechanics examples from the transcript
High jumper analogy: the jumper must apply a force to alter direction and speed as they approach the bar; the force applied depends on technique and required acceleration.
Block start (sprinter): to achieve rapid acceleration, you must apply large forces to the ground, which translates into higher GRF and a quicker increase in velocity.
Rehabilitation example: using lighter resistance bands can reduce the required force and acceleration, making controlled range of motion easier for patients recovering from injury. This aligns neuromuscular control with safer, slower movement while still achieving therapeutic goals.
Shopping cart analogy for inertia: empty cart is easier to accelerate than a loaded cart; same mass but different inertia due to mass distribution and total mass.
Pushing a wall example: equal and opposite forces exist (action and reaction), but the wall’s mass and the friction at the contact surface prevent noticeable acceleration of the wall; your body can still be propelled by the ground reaction force when force is applied with the feet.
Torque, rotation, and angular forces
Torque (rotational force): the effect of a force applied at some distance from an axis of rotation, producing rotation.
Magnitude: \tau = r F \sin\theta where $r$ is the lever arm length (distance from axis to point of force application) and $\theta$ is the angle between the force vector and the lever arm.
More generally, vector form: \boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}.
Units: newton-meters (N·m).
Moment arm: the distance from the axis of rotation to the line of action of the force; longer moment arms increase the rotational effect for a given force.
Musculoskeletal relevance: joint torques generated by muscles depend on both the force produced by the muscle and the moment arm available at the joint; training and rehabilitation often target altering either component to optimize movement or reduce injury risk.
Connections to foundational principles, real-world relevance, and ethics/practical implications
Foundational principles: Newton's laws, conservation of momentum, and action-reaction form the backbone of biomechanics and sports science.
Real-world relevance:
Gait mechanics (walking/running) rely on GRF patterns to quantify loading and efficiency.
Sprinting, jumping, and lifting depend on forces applied and the body's response through acceleration and velocity changes.
Rehabilitation and training programs adjust force and resistance to optimize motor learning while minimizing injury risk.
Ethical/practical implications:
When prescribing rehabilitation loads, consider safety, tolerance, and neuromuscular control; lighter resistance may enable safer progress and better control.
Understanding GRF helps in designing equipment, surfaces, and footwear to reduce injury risk and improve performance.
Accurate measurement and interpretation of forces can guide clinical decisions, return-to-sport timelines, and performance optimization.
Summary of key equations and units to memorize
Newton's second law (linear):
F = m a
Weight due to gravity:
W = m g, \quad g \approx 9.81\ \text{m/s}^2
Linear momentum:
\mathbf{p} = m \mathbf{v}
Work (force and displacement):
W = F \cdot d \cos \theta
Ground reaction force (conceptual): reaction to the force you apply to the ground; Newton's third law pair with the ground.
Torque (rotational force):
Magnitude: \tau = r F \sin \theta
Vector form: \boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}
Units: \text{N} \cdot \text{m}
Moment arm: distance from axis of rotation to line of action of the force; longer arms yield larger torques for the same force.
Key takeaways for exam-ready understanding
Unbalanced forces cause acceleration; inertia resists changes in motion; momentum quantifies resistance to changes in motion when moving.
Gravity contributes to weight and affects both vertical motion and the ground reaction force during movement; you must overcome gravity to change vertical motion.
Ground reaction force is central in gait and sports biomechanics; it reflects the interaction between the body and the ground and can be measured with force platforms.
Forces come in many flavors (gravitational, frictional, drag, spring, tension, applied); torque is the rotational counterpart to linear force and depends on the moment arm.
Clear distinction between static equilibrium (net force = 0) and dynamic movement (net force changes velocity; with enough force you can accelerate or decelerate objects or body segments).
Application to rehab and performance: adjust force/amount of resistance to control acceleration and ensure safe, progressive loading; use the concepts of GRF and torque to optimize movement patterns and reduce injury risk.
Quick review prompts
What is the difference between inertia and momentum? How does mass influence each?
How does the weight of a person relate to the force they must overcome to start moving upward?
How do friction, air resistance, and gravity collectively affect a moving ball on Earth versus in a vacuum?
How is ground reaction force related to the force you apply to the ground during running or jumping?
What determines the rotational effect of a force applied to a limb around a joint? How does the moment arm come into play?