Section 2: Forces and Newton's Laws
Section Learning Outcomes:
Section Key Terms:
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Lesson 1: Forces and Free-Body Diagrams
Free-Body Diagrams
A convenient way to analyse situations involving forces is to use diagrams. (FBDs) are visual representations used in biomechanics and physics to analyse the forces acting on an object.
Let's analyse the example of a person performing a bicep curl exercise to explain the concept of free-body diagrams. Imagine a person holding a dumbbell in their hand and performing a bicep curl, which involves flexing their elbow to lift the weight. To analyse the forces involved, we can create a free-body diagram that illustrates the forces acting on the person and the dumbbell. The free-body diagram of the bicep curl is shown in figure 1.
Figure 1: Free-body diagram of a bicep curl.
The object itself, such as the muscles involved in a bicep curl, can be represented by a dot or drawn as a small sketch. The directions and the approximate magnitudes of the forces are drawn on the diagram as arrows facing away from the object. A coordinate system is shown in the free-body diagram, with the +x and +y directions indicated.
In the example of the bicep curl, there are three types of forces involved. First is the weight of the dumbbell that acts vertically downward from the center of mass of the dumbbell. Second is the force that the person’s bicep muscle exerts to lift the dumbbell. Lastly, there are several reaction forces acting on the body such as the reaction force at the elbow joint due to the interaction between the forearm and the upper arm.
In solving word problems, especially complex ones, it is sometimes helpful to sketch a diagram of the system, called a system diagram, before drawing a free-body diagram. Following this with correctly drawing and labeling a free-body diagram is an important next step for solving a problem. It will help you visualize the problem and correctly apply the mathematics to solve the problem.
Common Forces
The concept of force lies at the heart of which is the study of how forces affect the motion of objects. In simple terms, a is the cause of motion, such as a push or a pull, which can lead to the acceleration, deceleration, or change of direction of an object. Newton's laws of motion are the foundation of dynamics and describe the way objects speed up, slow down, stay in motion, and interact with other objects.
Forces act upon objects by either pushing or pulling them. These objects can range from inanimate ones like tables to animate ones like people. The application of force can come from a person exerting it or even from the gravitational pull of Earth itself. Forces vary in both magnitude and direction, meaning some forces are stronger than others and can act in different orientations. For instance, the cannon in Figure 2 exerts a formidable force on a cannonball, propelling it high into the air. On the other hand, a mosquito landing on your arm applies a mere fraction of force in comparison, exerting only a slight pressure on your skin.
Figure 2: Cannon launching a cannon ball.
The , Fg, is the force that keeps our feet on the ground and planets in orbit around the Sun. It is the force of attraction between all objects and it is an action-at-a-distance force, which means that contact between the objects is not required. Essentially, it's like a cosmic magnet pulling objects towards each other. Imagine jumping off a diving board into a pool. Gravity is what brings you back down into the water, making it harder to do flips or stay in the air for too long, as seen in Figure 3.
Figure 3: Person jumping off a diving board into a pool with the force of gravity shown.
Gravity exists because material exists. However, the force of gravity is extremely small unless at least one of the objects is very large. For example, the force of gravity between a 1.0 kg ball and Earth at Earth’s surface is 9.8 N, but the force of gravity between two 1.0 kg balls separated by a distance of 1.0 m is only 6.7*10-11 N, a negligible amount.
Imagine holding one of the 1.0 kg balls in your hand. The force of gravity acts downward, toward Earth’s centre. However, to keep the ball stationary in your hand, there must be an upward force acting on it. This force, called the , FN, is like an invisible hand pushing back on objects to keep them from sinking through a surface and it is the force perpendicular to the two surfaces in contact. Without the normal force, things would simply fall through tables, chairs, and floors! Figure 4 shows the normal force acting vertically upward when the contact surfaces are horizontal.
Figure 4: A person holding a ball in their hand with the force of gravity and the normal force shown.
, T, is a fascinating concept that comes into play during a game of tug-of-war. Imagine two teams gripping a rope, each trying to pull the other side towards them. The tension force is what makes the game possible and determines which team emerges victorious. As the teams pull in opposite directions, the rope stretches and becomes taut. This tension force is transmitted through the rope, acting equally and in opposite directions on both teams. The more force each team exerts, the greater the tension in the rope becomes. It's the tension force that determines the outcome of the game, as the team that manages to exert a greater force eventually pulls the other team across a designated line.
Figure 5: Tug-of-war game with the tension forces shown.
An important characteristic of the tension force in a material such as the rope in a game of tug-of-war is that it has the same magnitude everywhere along the length of the material. This is true even if the direction of the force changes.
Another common force is , Ff, - the force that resists motion or attempted motion between objects in contact. Friction is a crucial concept that comes into play when riding a bicycle. Imagine yourself pedaling along a road. As your tires make contact with the ground, the force of friction opposes your motion and helps you maintain control, as seen in Figure 6. Friction occurs due to the interaction between the surfaces of the tires and the road. The friction force acts in the opposite direction to your bike's motion, which allows you to grip the road and propel yourself forward. Without friction, your tires would slip and slide, making it extremely challenging to maintain balance and move forward efficiently. However, friction also has its limitations. If you apply the brakes too suddenly, the friction between the brake pads and the bike's wheels increases, causing the bike to decelerate or come to a stop.
Figure 6: A person riding a bicycle with the friction force shown.
Finally, because there are several possible names for various pushes, pulls, thrusts, and so on, we will use the general term applied force, Fapp, for any contact force that does not fit any of the previously described categories.
Forces on Stationary Objects
When multiple forces exert their influence on an object, they combine to form the total force or . differ from , as they originate from outside the system and affect objects within it. On the other hand, internal forces act between two objects within the system. The encompasses both these concepts, representing the overall sum of external forces acting on the system.
The symbol for net force is ΣF, where the Greek letter sigma (Σ) serves as a reminder to add, or "sum", all the forces. Determining the sum of all the forces is straightforward if all the forces are linear or perpendicular to each other, but it is somewhat more complex if some forces are at angles other than 90 degrees. In two-dimensional situations, it is often convenient to analyze the components of the forces, in which case the symbols ΣFx and ΣFy are used instead of ΣF.
Lesson 2: Newton's Laws of Motion
Newton's First Law of Motion
Consider the air hockey table shown in Figure 9. When the air is turned off, the puck only slides a short distance before friction slows it to a stop. However, when the air is turned on, it lifts the puck slightly, so the puck experiences very little friction as it moves over the surface. With friction almost eliminated, the puck glides along with very little change in speed. On a frictionless surface, the puck would experience no net external force (ignoring air resistance, which is also a form of friction). This illustrates in action. Newton's first law of motion states the following: If the net force acting on an object is zero, that object maintains its state of rest or constant velocity.
Figure 9: A puck sliding on an air hockey table with the air off and on.
This law has important implications. An external net force is required to change an object’s velocity; internal forces have no effect on an object’s motion. For example, pushing on the dashboard of a car does not change the car’s velocity. To cause a change in velocity - in other words, to cause an acceleration - the net force acing on an object cannot be zero.
A common way to interpret this law is to say that an object at rest or moving with a constant velocity tends to maintain its state of rest or constant velocity unless acted upon by an external net force. The ability of an object to resist change to its motion is a fundamental property of all matter called . Inertia tends to keep a stationary object at rest or a moving object in motion in a straight line at a constant speed. Thus, the first law of motion is often called the law of inertia.
A helpful way to think about inertia is in terms of the object’s mass. Inertia is directly related to an object's mass: the greater the mass, the greater the inertia. For example, a sports car has a small inertia compared to a train. Thus, the car requires a much smaller net force than the train to cause it to accelerate from rest to a speed of 100 km/h. When the car and the train are travelling at the same speed, the train has a much larger inertia than the car.
Any object that has zero net force acting on it is in a state of equilibrium. In this sense, equilibrium is the property of an object experiencing no acceleration. The object can be at rest (static equilibrium) or moving at a constant velocity (dynamic equilibrium). In analysing the forces on objects in equilibrium, it is convenient to consider the components of the force vectors. In other words, the condition for equilibrium, ΣF=0, can be written as ΣFx=0 and ΣFy=0.
Newton's Second Law of Motion
is a fundamental principle that describes the relationship between force, mass, and acceleration. It states that if the external net force on an object is not zero, the object accelerates in the direction of the net force. The acceleration is directly proportional to the net force and inversely proportional to the object's mass.
To illustrate this law, let's consider the example of launching a rocket into space. When a rocket is ignited, it experiences a powerful force pushing it upward. According to Newton's second law, the acceleration of the rocket depends on both the magnitude of the force applied and the mass of the rocket itself.
If we compare two rockets with different masses but experience the same force, the lighter rocket will accelerate faster because it has less mass to overcome. On the other hand, a heavier rocket will accelerate at a slower rate due to its greater mass. This relationship between force, mass, and acceleration is mathematically expressed as
where F represents the net force in Newtons (N), m represents the mass in kg, and a represents the acceleration in m/s2.
Therefore, to achieve a higher acceleration and propel the rocket into space more quickly, engineers must either increase the force applied or reduce the rocket's mass. This law provides valuable insights for designing and optimizing various vehicles, such as rockets, cars, and airplanes.
Newton's second law has broad applications beyond space exploration. It helps explain the acceleration of athletes, the motion of vehicles, and even the impact of forces in everyday situations.
How does Newton's second law relate to his first law? According to the second law, a=ΣF/m, if the net force is zero, the acceleration must be zero, which implies that the velocity is constant (and could be zero). This agrees with the first law statement. It is evident that the first law is simply a special case of the second law, where ΣF=0.
Weight and Earth's Gravitational Field
We can apply Newton’s second law to understand the scientific meaning of weight. The of an object is the force exerted on an object due to gravity. Notice that this definition is different from , which is the quantity of matter of the object measured in kilograms (kg). The relationship between weight and mass is described by Newton’s second law of motion. It states that weight, Fg, is equal to the mass, m, of an object multiplied by the acceleration due to gravity, g. Mathematically, it can be expressed as
Near Earth, weight results from Earth's relatively large force of attraction on other bodies around it. The space surrounding an object in which a force exists is called a . The gravitational force field surrounding Earth extends from Earth’s surface far into space. At Earth's surface, the amount of force per unit mass, called the , is 9.8 N/kg [down].
Newton's Third Law of Motion
is an intriguing principle that states that for every action force, there is an equal reaction force in magnitude but opposite in direction. To illustrate this law, let's delve into an exciting example from the world of sports: kicking a soccer ball.
When a soccer player kicks a ball, they apply a force to the ball in one direction. As a result of Newton's third law, the ball exerts an equal and opposite reaction force on the player's foot. This reaction force is what allows the player to feel the impact of the kick.
Figure 13: Christine Sinclair kicking a football (soccer) ball (TSN).
Consider the scenario when a player strikes the ball with a powerful kick. The ball rapidly accelerates forward due to the force applied, but simultaneously, the player's foot experiences a backward reaction force. This reaction force prevents the foot from passing through the ball and allows the player to maintain balance.
In sports like martial arts, this principle is also evident. When a martial artist strikes an object, such as a punching bag, the force applied by the strike results in an equal and opposite reaction force exerted on their hand or body. This reaction force provides feedback to the martial artist, allowing them to adjust their technique and gauge the impact of their strikes.
Understanding Newton's third law in sports empowers athletes to enhance their performance. For example, in sports like tennis or table tennis, players can use the reaction force from their opponent's shot to generate a powerful counter-attack. They can also employ the principle to optimize their balance and stability during actions like jumping, running, or changing direction.
Applying Newton's Laws of Motion
One example of a situation in which forces are analyzed is when a car comes to a stop from a certain velocity. When a driver applies the brakes, several forces come into play. First, the driver's foot applies a force on the brake pedal. This force is transmitted through the brake system, causing the brake pads to press against the rotating discs or drums. The friction between the brake pads and the discs or drums generates a braking force.
By analyzing the forces at work, we can understand the physics behind the car's deceleration. The braking force opposes the car's motion, causing it to slow down. The magnitude of the braking force depends on factors such as the force applied by the driver, the effectiveness of the braking system, the condition of the tires, and the coefficient of friction between the brake pads and the discs.
Studying these forces allows engineers and scientists to design and optimize braking systems for safe and efficient vehicle stopping. By analyzing the forces involved, they can determine the necessary braking force to bring the car to a stop within a desired distance and time, ensuring the safety of the driver and passengers. Additionally, understanding the forces at play helps drivers anticipate and adjust their braking behavior in different road conditions, such as wet or icy surfaces, where the frictional forces may vary.
Newton's laws of motion can be used to solve a variety of problems. One approach to solving most problems involves following a series of steps.
- 1. Read the problem carefully and check the definitions of any unfamiliar words.
- 2. Draw a system diagram. Label all relevant information, including any numerical quantities given. For simple situtations, this step can be omitted.
- 3. Draw a free-body diagram of the object (or group of objects) and label all the forces. Choose the +x and +y directions. Try to choose one of these directions as the direction of the acceleration.
- 4. Calculate and label the x- and y-components of all the forces on the free-body diagram.
- 5. Write the equations of equilibrium using Newton's second law of motion: ΣFx=max and ΣFy=may, and substitute for the variables on both sides of the equations.
- 6. Repeat steps 3 to 5 for any other objects as required.
- 7. Solve the resulting equation(s) algebraically.
- 8. Check to see if your answers have appropriate units, a reasonable magnitude, a logical distance (if required), and correct number of significant figures.
Lesson 3: Exploring Frictional Forces
Coefficients of Friction
When it comes to the force of friction, there are two types to consider: static friction and kinetic friction. Let's explore both using the example of riding a bicycle.
comes into play when an object is at rest or trying to start moving. Imagine you are about to start pedaling on your bicycle from a standstill. Initially, the static friction force between your bike's tires and the road prevents them from slipping and allows you to push off and start moving. It's like a helpful grip that keeps your tires in place until you exert enough force to overcome it.
Once you start pedaling and your bicycle is in motion, the force of friction changes to . Kinetic friction acts when there is relative motion between the surfaces in contact. In the case of riding a bicycle, kinetic friction occurs between the tires and the road as they continue to roll. Kinetic friction helps maintain traction, allowing you to steer, control your speed, and navigate turns.
It's important to note that the force of kinetic friction is typically slightly lower than static friction, which means it requires less force to keep an object in motion than to initially set it in motion. This is why it can be easier to maintain your speed while riding a bicycle rather than getting it started from a complete stop.
Figure 18: Graph depicting the magnitude of friction as a function of the magnitude of the applied force on an object up to the instant the object begins to move.
The magnitudes of the forces of static and kinetic friction depend on the surfaces in contact with each other. For example, a fried egg in a non-stick frying pan experiences little friction, whereas a sleigh pulled across a concrete sidewalk experiences a lot of friction. The magnitude of the force of friction also depends on the normal force between the objects.
The coefficient of friction is a number that indicates the ratio of the magnitude of the force of friction between two surfaces to the normal force between those surfaces. The value for the coefficient of friction depends on the nature of the two surfaces in contact and the type of friction - static or kinetic. The , μS, is the ratio of the magnitude static friction to the magnitude of the normal force.
The , μK, is the ratio of the magnitude of the kinetic friction to the magnitude of the normal force. The corresponding equations are
Determining μS and μK for given substances is done empirically, or through experimentation. Results of such experiments may differ from one laboratory to another, even with careful measurements and sophisticated equipment. For example, if several scientists at different locations in Canada measure the coefficient of kinetic friction between wood and dry snow, the wood and snow samples would vary; therefore, the coefficient values would not be consistent.
Lesson 4: Inertial and Non-Inertial Frames of Reference
Inertial and Non-Inertial Frames of Reference
In the context of an astronaut floating freely in space, we can explore the concepts of inertial and non-inertial frames of reference.
An is a coordinate system in which Newton's laws of motion hold true without any additional forces or acceleration acting on objects within that frame. In the scenario of an astronaut floating freely in space, far away from any significant gravitational fields or external forces, the astronaut's frame of reference can be considered inertial. From the perspective of the astronaut, objects and the astronaut's own body move with a constant velocity or remain at rest unless acted upon by external forces.
Figure 21: An astronaut floating in space.
In contrast, a is one in which additional forces or acceleration are present, making the motion of objects within that frame more complex. For example, if the astronaut were inside a spacecraft that is accelerating or decelerating, the frame of reference within the spacecraft would be non-inertial. In this case, the astronaut would experience the sensation of being pushed backward during acceleration or forward during deceleration. The additional forces due to the spacecraft's acceleration or deceleration would make the motion within the frame non-inertial.
Figure 22: An astronaut in a spacecraft accelerating.
The distinction between inertial and non-inertial frames of reference is important because it helps us analyze and understand the motion of objects accurately. In an inertial frame, objects behave according to the laws of physics, and their motion can be predicted using Newton's laws. In a non-inertial frame, additional forces or acceleration must be considered to account for the apparent motion of objects.