Linear and Angular Motion
Kinematics
Kinematics is the branch of mechanics that studies motion without considering the forces that cause it. It focuses on describing motion using parameters such as position, displacement, speed, velocity, and acceleration.
Motion
Motion is the change in position of an object or body over time. It can involve movement from one location to another or changes in the position of individual body parts relative to each other. Motion is described using key concepts such as displacement, velocity, and acceleration.
Motion can be:
- Linear (motion in a straight line), such as a sprinter running down a track.
- Curvilinear (motion along a curved path), such as a football being kicked in a long arc.
- Angular or rotational (motion around an axis), such as a figure skater spinning on one foot.
- General (a combination of linear and angular motion), such as a cyclist pedaling while moving forward.
Linear Motion
Linear motion refers to movement along a straight path.
It can be described using three key terms:
- Speed: The distance covered per unit of time. It's a scalar quantity, meaning it has no direction.
- Velocity: The rate of change of displacement (distance in a specific direction) over time. It's a vector quantity, meaning it includes both magnitude and direction.
- Acceleration: The rate of change of velocity over time. It can be positive (speeding up) or negative (slowing down).
- A sprinter running 100 meters in 10 seconds has a speed of 10 m/s.
- If they start from rest and reach this speed in 2 seconds, their acceleration is 5 m/s².
Measurements and Position
Position
Position refers to the location of an object or body, typically given by its coordinates.
Coordinates
Coordinates measure distance from an origin (e.g., in meters) and are given in two dimensions (x, y: horizontal and vertical) or three dimensions (x, y, z: horizontal, vertical, lateral). Two systems for three-dimensional coordinates: System 1: x = horizontal, y = vertical, z = lateral and System 2: x = horizontal, y = lateral, z = vertical. Angular coordinates are measured as angles around one or more axes.
Newton's laws of motion and linear motion
1. Newton's first law: the law of inertia
Newton's First Law
An object will remain at rest or keep moving at a constant velocity unless acted upon by an outside force. This law is also known as the law of inertia
- This law emphasizes that a force is necessary to change the motion of an object.
- This law explains why it is harder to start or stop an object's motion.
- The resistance to change in motion is called inertia.
Inertia
Inertia is a property of matter that causes an object to resist changes in its state of motion. The more mass an object has, the greater its inertia.
- A soccer ball will not move unless kicked (object at rest) and will continue to roll in a straight line unless acted upon by external forces like friction or a player kicking it again (object in motion).
- A hockey puck glides across the ice until friction or a player's stick changes its motion.
Application in Sports
- Inertia explains the challenge faced by athletes when starting and stopping motion.
- It takes significant force to overcome inertia and accelerate a stationary object or decelerate a moving object.
- Running: A sprinter at the starting line has to apply force to overcome inertia and accelerate from rest. The sprinter’s body tends to stay at rest until sufficient force is applied.
- Football: A football, once kicked, continues in motion until external forces (e.g., friction with the ground, air resistance) slow it down or stop it.
- Stopping: It takes more effort to stop a larger object (like a heavy sled) compared to a smaller one because of inertia.
This law explains why athletes need to exert more force to overcome inertia when starting or stopping movement.
2. Newton's second law: the law of acceleration
Newton's Second Law
The force on an object is equal to its mass times its acceleration. This law states that the greater the force applied to an object, the greater the acceleration.
The acceleration of an object is proportional to the force applied and inversely proportional to its mass.
This is expressed as: F = ma
Where:
- F is the force acting on the object (measured in Newtons, N),
- m is the mass of the object (measured in kilograms, kg),
- a is the acceleration of the object (measured in meters per second squared, m/s²).
- This law shows how the force applied to an object influences its motion.
- The greater the force applied to an object, the greater its acceleration, provided that the mass remains constant.
- Students often think more force always means more speed.
- Remember - the object's mass matters!
- A heavy shot put needs much more force than a tennis ball for the same acceleration.
- Acceleration is change in velocity demanded by the time taken.
- So Newton's second law of motion could be rewritten as:
$$F = \frac{m(v - u)}{t}$$
Application in Sports
- To maximize acceleration, athletes aim to increase the force they apply and/or reduce their mass.
- For example, a sprinter can work on strengthening their legs (to increase the force they apply to the ground) or reduce body fat to lower their mass.
Sprinting
- A sprinter needs to apply a large force to the ground to generate a high acceleration from the starting blocks.
- The larger the force they exert, the quicker they can accelerate.
Javelin Throw
- The athlete applies a large force to the javelin, which accelerates it through the air.
- The mass of the javelin limits how much acceleration can be achieved for a given force.
- When working through problems involving acceleration, always look at the force and mass variables.
- If mass increases, for the same force, acceleration will decrease.
- If force increases, acceleration increases (as long as mass remains constant).
3. Newton's third law: the law of action-reaction
Newton's Third Law
When two objects interact, they apply forces to each other of equal magnitude and opposite direction. This law is also known as the law of action and reaction
- The Third Law of Motion states that for every action, there is an equal and opposite reaction.
- The action force is the force that is exerted by one object, and the reaction force is the equal and opposite force exerted by the second object.
- This means that when one object applies a force on another object, the second object applies an equal but opposite force on the first object.
- The action and reaction forces are always equal in magnitude and opposite in direction, but they act on different objects.
- Remember that action and reaction forces act on different objects.
- For example, when jumping, the action force is applied to the ground, and the reaction force acts on the athlete's body.
Application in Sports
- While action and reaction forces are equal in size, they do not cancel out because they act on different objects.
- The force an athlete applies to the ground results in an opposite force that propels them in the opposite direction.
- Jumping: When an athlete pushes off the ground (action), the ground pushes back with an equal force, propelling the athlete upward (reaction).
- Swimming: A swimmer pushes water backward with their hands and feet (action), and the water pushes the swimmer forward (reaction).
- Walking: When a person walks, their foot pushes backward against the ground (action), and the ground pushes them forward (reaction).
- Students may incorrectly assume that action and reaction forces cancel each other out.
- These forces do not cancel because they act on different objects.
Key principles of Linear motion
Stability
The ability to maintain equilibrium and resist changes in position, determined by several factors affecting balance.
Four key factors influence stability in sport:
- Height of center of mass (lower = more stable)
- Size of base of support (wider = more stable)
- Position of line of gravity (should fall within base)
- Total mass (more = greater stability)
Wrestlers demonstrate all stability principles by:
- Lowering their center of mass
- Widening their stance
- Keeping their line of gravity centered
- Using their body mass effectively
Don't confuse stability with balance! Stability is resistance to movement, while balance is maintaining controlled position. A sumo wrestler is very stable but might not have great balance for gymnastics.
Distance and Displacement
Distance
Distance
Distance is the total length of the path traveled by an object, regardless of direction.
Distance is a scalar quantity, meaning it only has magnitude (size) and no direction.
Distance=Total Path Traveled
ExampleIf a runner completes two laps around a 400 m track, the total distance covered is 800 m.
Displacement
Displacement
Displacement is the shortest straight-line distance between an object’s starting and ending positions, including direction.
Displacement is a vector quantity, meaning it has both magnitude and direction.
ExampleIf a runner starts at point A, runs 100 m to point B, then returns 100 m back to point A:
- Distance traveled = 200 m
- Displacement = 0 m (since the runner ended at the starting position)
Think of distance as the number of steps you take during a walk, while displacement is how far you are from where you started.
Linear Speed, Velocity, and Acceleration
Speed
Speed
Speed is the rate of change of distance over time.
- It is a scalar quantity, meaning it only has magnitude and no direction.
- It is measured in meters per second (m/s).
$$ \text{Speed} = \frac{\text{Distance}}{\text{Time}} $$
ExampleA runner covering 100 meters in 10 seconds has a speed of 10 m/s.
NoteSpeed does not indicate the direction of motion, only how fast an object is moving.
Velocity
Velocity
Velocity is the rate of change of displacement and includes direction.
- Unlike speed, velocity is a vector quantity (it has both magnitude and direction).
- Velocity is measure in meters per second (m/s)
- Velocity can be positive, negative, or zero depending on direction.
$$ \text{Velocity} = \frac{\text{Displacement}}{\text{Time}} $$
ExampleIf a cyclist moves 100 m east in 10 s, the velocity is 10 m/s east.
Acceleration
Acceleration
Acceleration is the rate of change of velocity over time.
- It describes how quickly an object speeds up, slows down, or changes direction.
- The unit is meters per second squared (m/s²)
$$ \text{Acceleration} = \frac{\text{Change in Velocity}}{\text{Time Taken}} $$
Types of acceleration:
- Positive acceleration: Speeding up (e.g., a runner increasing speed).
- Negative acceleration (deceleration): Slowing down (e.g., a runner stopping after a sprint).
A sprinter increases velocity from 0 to 8 m/s in 2 seconds.
Relationship Between Force, Acceleration, and Mass
Newton’s Second Law states that the force acting on an object is equal to the product of its mass and acceleration:
F=ma
Where:
- F = Force (Newtons, N)
- m = Mass (kg)
- a = Acceleration (m/s²)
- If mass remains constant, increasing force increases acceleration.
- If force remains constant, increasing mass decreases acceleration.
- A heavier rugby player requires more force to accelerate than a lighter player.
- A baseball pitcher applies more force to throw a fastball than a slow pitch.
- Think of pushing a shopping cart.
- If the cart is empty, it is easy to accelerate.
- If the cart is full of groceries, you need to apply more force to move it at the same acceleration.
- Acceleration occurs whenever velocity changes, even if the speed stays the same.
- If an object moves in a circle at constant speed, it is still accelerating due to the change in direction.
Impulse and momentum
Momentum
Momentum is the product of mass and velocity.
p = m*v
where:
- p=momentum (kgm/s)
- m=mass (kg) and
- v= velocity (m/s)
Impulse
Impulse is the product of force and the time.
Impulse causes a change in momentum, as described by the impulse- momentum relationship:
Change in momentum = Impulse = J=F×Δt
where:
- J=Impulse (Ns)
- F= Force applied (N) and
- Δt change in time (s)
The impulse-momentum relationship is crucial in sports:
- Longer force application = greater change in momentum
- Direction of force determines direction of motion
- Impact forces can be managed through time of application
Think of landing from a jump.
- Hard, stiff landing = short time, high force.
- Soft, bent-knee landing = longer time, lower force.
- Same impulse, different force experience!
Angular Motion
Angular Motion
The motion of a body about a fixed point or fixed axis
- Angular motion occurs when a force is applied at a distance from the axis of rotation, causing the object to rotate instead of moving in a straight line.
- Angular motion involves rotation around an axis, such as a gymnast spinning or a diver twisting.
Characteristics of Angular Motion
- It takes place around a fixed axis, which can be real (e.g., bicycle wheel) or imaginary (e.g., gymnast in mid-air).
- All points on the object move in circular paths around the axis.
- Different points on the object move at different linear speeds, but they share the same angular displacement, angular velocity, and angular acceleration.
- The radius of the rotation affects how fast different parts move: the further from the axis, the greater the linear speed, even if the angular speed remains the same.
Understanding angular motion is essential for sports involving rotation like diving, gymnastics, and figure skating.
- A gymnast performing a somersault rotates around their center of mass.
- A diver executing a twisting dive rotates around their longitudinal axis.
- A basketball player spinning the ball on their finger applies angular motion.
- A cyclist’s wheels exhibit angular motion while moving forward in linear motion.
Angular Displacement (θ)
Angular displacement
Angular displacement refers to the change in the angular position of a rotating body.
- Angular displacement is a vector quantity, meaning it has both magnitude and direction.
- It can be positive (counterclockwise) or negative (clockwise) based on the right-hand rule.
- It differs from angular distance, angular displacement measures the shortest path, while angular distance includes all rotations.
- A gymnast performs a backflip, rotating 270° before landing.
- A diver rotating 1.5 turns before entering the water completes an angular displacement of 540° (1.5 × 360°).
Angular velocity
Angular velocity
Angular velocity measures the rate at which an object rotates around an axis.
- Angular velocity is a vector quantity and has both magnitude (speed of rotation) and direction.
- The direction of angular velocity follows the right-hand rule.
- If the rotation is counterclockwise, the angular velocity is positive, if it is clockwise, the angular velocity is negative.
- A tennis racket swing has high angular velocity, meaning it rotates quickly.
- A basketball player spinning the ball on their finger applies angular motion to the ball.
It is given by:
$$ \text{ω} = \frac{\text{θ}}{\text{t}} $$
Where:
- ω= Angular velocity (rad/s)
- θ= Angular displacement (rad)
- t= Time taken (s)
- Angular velocity is similar to linear velocity, but instead of describing how fast an object moves in a straight line, it describes how fast the object rotates.
- Think of the difference as a car moving forward (linear velocity) versus a car spinning on its wheels (angular velocity).
Relationship between linear and angular velocity
v=r⋅ω
Where:
- v = Linear velocity (m/s)
- r = Radius (m)
- ω = Angular velocity (rad/s)
Angular acceleration
Angular acceelration
Angular acceleration is the rate at which angular velocity changes over time.
It is measured in radians per second squared (rad/s²)
It is given by:
$$α= \frac{Δω}{Δt}$$
where:
- α= Angular acceleration (rad/s²)
- Δω= Change in angular velocity (rad/s)
- Δt = Time interval (s)
- If α>0, the object is speeding up (positive acceleration).
- If α<0, the object is slowing down (negative acceleration or deceleration).
- A tennis player’s arm accelerates during a serve.
- A basketball spinning on a finger slows down due to friction, experiencing negative angular acceleration.
Introduction to Kinematics
Force
A force is a mechanical interaction between two objects or bodies.
Resultant Motion
Determined by the sum of all forces acting on an object. Direction and magnitude of forces influence height and speed. Adjusting technique can optimize trampoline force and minimize gravity’s effect.
Gravity
An attractive force that acts between objects with mass. Proportional to mass, inversely proportional to the square of the distance. Effects of Gravity: Causes planetary orbits and is responsible for the Earth’s attraction to objects near its surface.
Mass and Weight
Mass
The amount of material in a body or object.
Weight
The gravitational force acting on an object due to a gravitational field.
Instantaneous vs. Average Kinematics
Instantaneous Velocity
Velocity measured over an extremely short time interval ("instant").
Instantaneous acceleration
Instantaneous acceleration is the rate of change of velocity at a specific instant in time.
Angular Momentum
Angular Momentum
Angular momentum measures the amount of rotational motion an object has or can achieve. It is generated by torque when an eccentric force is applied to an object that is free to rotate around an axis. This force may come from external objects (e.g., the ground or sports equipment), another body, or muscle movements. Angular momentum is a vector quantity, meaning it has both size and direction, and it is measured in kg·m²/s. It is represented by the letter L and is given by the equation: L=Iω where I is the moment of inertia and ω is the angular velocity.
Conservation of Angular Momentum
Conservation of angular momentum
Conservation of angular momentum is a physical property of a spinning system such that its spin remains constant unless it is acted upon by an external torque
- Angular momentum remains constant unless acted upon by an external torque.
- This principle explains why a diver can control their spin by tucking or extending their body mid-air.
Angular momentum only changes when the gymnast is in contact with the ground or the box and remains constant during airborne phases.
Phase Breakdown:
- Phase A: The gymnast is in contact with the ground, generating angular momentum in the clockwise direction, exceeding 60 kg·m²/s.
- Phases B & D: The gymnast is airborne (not in contact with the ground or box), so angular momentum remains constant.
- Phase C: The gymnast’s arms make contact with the box, causing a decrease in angular momentum as their angular velocity is reduced.
- Phase E: The gymnast lands on the ground, and angular momentum decreases to zero.
Transfer of Angular Momentum
- Since angular momentum cannot be created or destroyed unless an external torque is applied, an athlete cannot change their total angular momentum once airborne.
- Internal Adjustments in Angular Velocity
- When one part of the body increases its angular velocity (e.g., through muscle contraction), another part must decrease its angular velocity to maintain total angular momentum.
- When the first part slows down, the second part speeds up again, ensuring conservation of momentum.
- In a piked dive, the diver flexes their hips to assume a tucked position (0–0.5 seconds).
- This increases the angular velocity and momentum of the upper body.
- To maintain total angular momentum, the lower body’s angular momentum decreases simultaneously.
- The greater the upper body's angular momentum, the lower it is for the lower body, balancing the motion.
Trading Angular Momentum
- Since angular momentum is a vector quantity, an athlete rotating about one axis (e.g., during a somersault) can introduce rotation around another axis (e.g., tilting) by adjusting body segment movements using muscles.
- When this happens, the combination of these two vectors results in rotation around a third axis, creating twisting motion. In gymnastics, this principle is used to generate twists during somersaults.
- In a Kasamatsu vault, a gymnast initially performs a handspring onto the vaulting table, followed by a somersault with an added twist in the air. By tilting their body slightly upon takeoff, they convert some of their somersault angular momentum into twisting motion, allowing them to execute a 1.5 or 2.5 twist before landing.
- This principle applies to other sports as well, such as diving (e.g., twisting dives) and trampoline gymnastics, where athletes manipulate their body positioning mid-air to control rotational movement.
Stability in Motion
- Stability is crucial in sports and depends on:
- Height of the Center of Mass: Lowering the center of mass increases stability.
- Size of the Base of Support: A wider base provides greater stability.
- Position of the Line of Gravity: Stability is maximized when the line of gravity falls within the base of support.
- Mass: Greater mass increases stability.
Wrestlers lower their center of mass and widen their stance to resist being toppled.
NoteDon't confuse stability with balance. Stability refers to resistance against being moved, while balance is the ability to maintain control over the body's position.
Summing Joint Forces
- In complex movements, forces generated by different joints combine to produce powerful actions.
- For example, in a volleyball spike, the legs, hips, and arms work together to maximize force and height.
- This is why it is important to use the adequate technique in order to avoid injuries if extra pressure is applied to joints
A high jumper uses their legs to generate upward force while their arms and torso contribute to the jump's height and direction.
Applying Newton's Laws to Sports
- Running: Sprinters use Newton's third law to push off the ground, while their acceleration is governed by the second law.
- Gymnastics: Angular momentum is conserved during flips, allowing gymnasts to control their rotation.
- Basketball: Players maintain stability by keeping their center of mass low and within their base of support.
How do cultural differences in sports training reflect the application of Newton's laws? For example, consider the emphasis on balance in martial arts versus power in weightlifting.
Self reviewA soccer player runs 10 m north, then 10 m south. What are the player’s total distance and displacement?
Principle of linear momentum and linear impulse
Linear Momentum
Property of an object due to its movement. Formula: p=mv, where: mm = mass (kg) and vv = velocity (m/s). Vector quantity (has both size and direction). Measured in kg·m/s.
Linear Impulse
Force applied over a period of time. Formula: J=FΔt where: FF = force (N) and ΔtΔt = time duration (s). Also a vector quantity.
Principle of impulse direction
Principle of Impulse Direction
In sports biomechanics, impulse direction refers to the way force is applied to an object to control its movement. For example, in soccer, the direction and magnitude of a kick determine where the ball goes. Impulse acts as a "pushing force," and its direction is crucial in guiding the ball’s trajectory.
Soccer Pass vs. Shot on Goal:
- A player kicks a soccer ball with a controlled pass along the ground by applying force in a straight direction.
- If they apply force at an angle or with more power, the ball lifts into the air or curves.
Principle of angular movement
Moment of Inertia
The moment of inertia measures how difficult it is for an object or body to rotate around an axis. It is expressed in kg·m² and depends on two key factors: Mass of the object or body and Distribution of mass relative to the axis of rotation. Greater mass farther from the axis increases the moment of inertia, making rotation more difficult. Mass closer to the axis decreases the moment of inertia, making rotation easier. The human body’s moment of inertia varies depending on body position and axis of rotation.
Torque
Torque, or the moment of force, occurs when a force is applied to an object that can rotate around an axis. This force, called an eccentric force, does not act directly through the axis but instead causes rotation. The amount of torque generated depends on: size of the force, direction of the force and the distance from the axis of rotation.
Cycling:
- Pedaling harder increases torque on the gears, making the bike accelerate faster.
- Adjusting the gear ratio changes the torque applied to the wheels.
Why do divers tuck their legs to spin faster?
