Chapter 2: Force - Comprehensive Notes

2.1 Interactions and Forces

• Mechanics: The branch of physics that deals with the study of motion and the forces that cause it. It is a fundamental area of physics that explains how objects move and interact. This includes kinematics (the description of motion) and dynamics (the study of the causes of motion).

• Interaction: Mutual or reciprocal action or influence. In physics, interactions are described and measured via two forces. Each force is exerted on one of the interacting objects, demonstrating Newton's Third Law of Motion.

• Force: A push or a pull exerted on an object. Forces can cause an object to accelerate, decelerate, change direction, or deform. They are vector quantities, possessing both magnitude and direction.

2.2 Long-range Forces

• Long-range forces: Forces that can act over a distance without physical contact between objects. These forces are mediated by fields.

• Gravity: An example of a long-range force. It is the force of attraction between any two objects with mass. The strength of the gravitational force depends on the masses of the objects and the distance between them.

• Weight: The magnitude of the gravitational force exerted by a planet or moon on an object. Weight is a force and is measured in Newtons (N) or pounds (lb).

2.3 Contact Forces

• Contact forces: Forces that occur when objects are in physical contact with each other. These forces arise from the interactions between atoms and molecules at the surfaces of the objects.

• The concept of "contact" is a simplification at the macroscopic level. At the atomic level, defining contact is problematic due to the nature of atomic interactions. Atoms do not actually "touch" but interact through electromagnetic forces.

2.4 Measuring Forces

• Spring Scale: A common device used to measure applied force. It operates by calibrating the stretch of a spring when a force is applied.

• The amount of stretch is proportional to the magnitude of the applied force, following Hooke's Law. The scale is calibrated to display the force in appropriate units.

2.5 Units of Force

• Pound (lb): A common unit of force in the United States customary system.

• Newton (N): The SI unit of force. It is defined as the force required to accelerate a mass of 1 kilogram at a rate of 1 meter per second squared ($1 \text{ N} = 1 \text{ kg} \cdot \text{m/s}^2$).

• Conversion:

  • $1 \text{ lb} = 4.448 \text{ N}$

  • $1 \text{ N} = 0.2248 \text{ lb}$

2.6 Force as a Vector Quantity

• Force is a vector: It has both magnitude and direction. This means that to fully describe a force, you must specify how strong it is (magnitude) and in what direction it is acting.

• Direction is as crucial as magnitude in describing a force. The direction determines the effect of the force on the object.

2.7 Scalars and Vectors

• Scalar Quantity: A quantity that has magnitude, algebraic sign, and units, but no direction in space (e.g., mass, temperature, time).

• Vector Quantity: A quantity that has both magnitude and direction (e.g., force, velocity, displacement).

• Scalar Addition: Scalars add algebraically. For example, if you have 3 kg of apples and add 2 kg more, you have 5 kg of apples ($3 \text{ kg} + 2 \text{ kg} = 5 \text{ kg}$).

• Vector Addition: Requires considering the directions of the vectors. Vectors can add constructively or destructively depending on their directions.

2.8 Vector Direction Matters

• The effect of multiple forces depends on their directions. Forces acting in the same direction combine to produce a larger effect, while forces acting in opposite directions can cancel each other out.

• Forces in the same direction add constructively, resulting in a larger net force.

• Forces in opposite directions diminish each other; if they are equal in magnitude, they can completely cancel out, resulting in zero net force.

2.9 Vector Notation

• Notation:

  • $\vec{F}$: Vector quantity (with arrow over boldface symbol). This notation is used to indicate that the quantity is a vector, having both magnitude and direction.

  • $F$: Magnitude of the vector (italicized). The magnitude is a scalar quantity representing the length or strength of the vector.

  • $\lVert\vec{F}\rVert$: Magnitude of the vector using absolute value bars. This notation emphasizes that only the magnitude of the vector is being considered.

• Magnitude: Always positive or zero; has units. The magnitude represents the size or strength of the vector and cannot be negative.

2.10 Graphical Vector Addition

• Vectors can be added graphically by placing the tail of one vector at the tip of another. This method is also known as the head-to-tail method.

• The resultant vector goes from the tail of the first vector to the tip of the last. The resultant vector represents the sum of all the vectors.

• Vector addition is commutative:
  $\vec{F}<em>1 + \vec{F}</em>2 = \vec{F}<em>2 + \vec{F}</em>1$

• Incorrect vector addition: Joining vectors tip-to-tip without proper orientation does not yield the correct resultant vector. Vectors must be placed head to tail.

• Examples:

  • 300 N + 200 N in the same direction = 500 N

  • 300 N - 400 N in opposite directions = 100 N (in the direction of the 400 N force)

  • 250 N + 250 N in opposing directions = 0 (sum is zero)

2.11 Vector Addition Using Components

• Any vector can be expressed as the sum of vectors parallel to the x-, y-, and z-axes. This is a fundamental concept in vector analysis, allowing complex vector problems to be broken down into simpler components.

• Components indicate the magnitude and direction along each axis. These components are scalar quantities and can be positive or negative, indicating direction along the axis.

• A component has magnitude, units, and an algebraic sign. The sign indicates the direction of the component along the axis.

• Resolving a vector: The process of finding its components. This is done using trigonometric functions.

2.12 Components of a Vector

• A vector can be resolved into perpendicular components using a two-dimensional coordinate system. This simplifies vector addition and analysis.

• Length, angle, & components are calculated using trigonometry. Trigonometric functions (sine, cosine, tangent) relate the angles and side lengths of right triangles.

• Signs of vector components depend on the quadrant. The quadrant in which the vector lies determines the signs of its x- and y-components.

2.13 Component Vectors

• A vector $\vec{A}$ can be represented as the sum of its x-component vector $\vec{A}<em>x$ and y-component vector $\vec{A}</em>y$: $\vec{A} = \vec{A}<em>x + \vec{A}</em>y$

• The x-component vector is parallel to the x-axis, and the y-component vector is parallel to the y-axis. These component vectors are perpendicular to each other.

2.14 Working with Components

• Vectors can be added using components. This method involves breaking down each vector into its x- and y-components and then adding the corresponding components.

• If $\vec{C} = \vec{A} + \vec{B}$, then $C<em>x = A</em>x + B<em>x$ and $C</em>y = A<em>y + B</em>y$. This shows how the components of the resultant vector $\vec{C}$ are found by adding the components of vectors $\vec{A}$ and $\vec{B}$.

2.15 Adding and Subtracting Vectors

• Adding vectors using components:

  1. Find the components of each vector.

  2. Add the x- and y-components separately.

  3. Find the resultant vector from the sums of the x- and y-components. This can be done graphically or using the Pythagorean theorem and trigonometric functions.

2.16 Adding and Subtracting Vectors

• Subtracting vectors: Adding the negative of a vector. This is equivalent to changing the direction of the vector and then adding it.

• The negative of a vector has the same magnitude but points in the opposite direction. If $\vec{A}$ is a vector, then $-\vec{A}$ has the same magnitude as $\vec{A}$ but points in the opposite direction.

2.17 Unit Vectors

• Unit vectors: Dimensionless vectors with a unit length. They are used to specify direction.

• Multiplying unit vectors by scalars: The multiplier changes the length, and the sign indicates the direction. For example, if $\hat{i}$ is a unit vector in the x-direction, then $5\hat{i}$ is a vector of length 5 in the x-direction, and $-3\hat{i}$ is a vector of length 3 in the negative x-direction.

2.18 Position, Displacement, Velocity, and Acceleration Vectors

• Position vector: Points from the origin to the location in question. It describes the location of a point in space.

• Displacement vector: Points from the initial position to the final position. It describes the change in position of an object.

2.19 Accelerated Motion on a Ramp

• Component of free-fall acceleration accelerates a crate down an incline. The component of gravity parallel to the ramp causes the acceleration.

• $a_x = \pm g \sin{\theta}$, where $\theta$ is the angle of the incline. This equation gives the acceleration along the ramp, with the sign depending on the direction.

2.20 Finding Vector Components

• To find the x- and y-components of a vector, draw a right triangle, determine the angle, and use trigonometric functions. This involves breaking down the vector into its horizontal and vertical components.

• Example: Force vector $\vec{F}$ with magnitude 9.4 N, directed 58° below the +x-axis. The x-component will be $9.4 \cos(58^\circ)$ and the y-component will be $-9.4 \sin(58^\circ)$.

2.21 Right Triangle Review

• For a right triangle:

  • $\sin{\theta} = \frac{\text{opposite}}{\text{hypotenuse}}$

  • $\cos{\theta} = \frac{\text{adjacent}}{\text{hypotenuse}}$

  • $\tan{\theta} = \frac{\text{opposite}}{\text{adjacent}}$

2.22 Problem-Solving Strategy: Finding Components

1. Draw a right triangle with the vector as the hypotenuse. This helps visualize the components.

2. Determine one of the angles in the triangle. This angle is used to find the components.

3. Use trigonometric functions to find the magnitudes of the components. Sine and cosine are used to find the components.

4. Determine the correct algebraic sign for each component. The sign depends on the quadrant in which the vector lies.

2.23 Finding Magnitude and Direction

• A vector's magnitude and direction can be found from its components. This is the reverse of finding components from magnitude and direction.

• $\lVert\vec{F}\rVert = \sqrt{F<em>x^2 + F</em>y^2}$

• $\theta = \tan^{-1}{\frac{F<em>y}{F</em>x}}$

2.24 Problem-Solving Strategy: Finding Magnitude and Direction

1. Sketch the vector on x- and y-axes. This helps visualize the vector and its components.

2. Draw a right triangle with the vector as the hypotenuse. This relates the components to the magnitude and direction.

3. Choose the unknown angle to determine. This angle will define the direction of the vector.

4. Use the inverse tangent function to find the angle. The inverse tangent function gives the angle whose tangent is the ratio of the y-component to the x-component.

5. Interpret the angle (below horizontal, west of south, etc.). The angle must be interpreted in the correct quadrant.

6. Use the Pythagorean theorem to find the magnitude of the vector. The Pythagorean theorem relates the magnitude to the components.

2.25 Example 2.3

• Deltoid muscle exerts a force of 270 N at 15° above the horizontal on the humerus.

• To find the x- and y-components:

  • $F_x = F \cos{\theta} = 270 \text{ N} \times \cos{15^\circ} = 260 \text{ N}$

  • $F_y = F \sin{\theta} = 270 \text{ N} \times \sin{15^\circ} = 70 \text{ N}$

• Check. $\sqrt{(-260 \text{ N})^2 + (70 \text{ N})^2} = 270 \text{ N}$

• $\theta = \tan^{-1}{\frac{70 \text{ N}}{260 \text{ N}}} = 15^\circ$

2.26 Adding Vectors Using Components

• If $\vec{C} = \vec{A} + \vec{B}$, then $C<em>x = A</em>x + B<em>x$ and $C</em>y = A<em>y + B</em>y$

2.27 Problem-Solving Strategy: Adding Vectors Using Components

1. Find the x- and y-components of each vector.

2. Add the x-components to find the x-component of the sum.

3. Add the y-components to find the y-component of the sum.

4. If necessary, find the magnitude and direction of the sum. Use the Pythagorean theorem and the inverse tangent function.

2.28 Net Force

• Net force is the vector sum of all forces acting on an object. It is the total force that determines the object's acceleration.

• If $\vec{F}<em>1, \vec{F}</em>2, \dots$ are all forces on an object, then the net force is: $\vec{F}<em>{\text{net}} = \vec{F}</em>1 + \vec{F}_2 + \dots$

2.29 Newton’s First Law

• Newton’s First Law of Motion: An object’s velocity vector remains constant if and only if the net force acting on the object is zero. This law is also known as the law of inertia.

2.30 Inertia

• Inertia: Resistance to changes in velocity. It is a measure of an object's tendency to stay in its current state of motion.

• Does not mean resistance to the continuation of motion or the tendency to come to rest. Inertia is not a force but rather a property of matter.

2.31 Free-Body Diagrams

• Free-body diagram (FBD): A simplified sketch of a single object with force vectors representing every force acting on it. FBDs are used to analyze the forces acting on an object.

• Must not include forces acting on other objects. The FBD should only show forces acting on the object of interest.

2.32 How to draw a Free-Body Diagrams

1. Draw the object in a simplified way. Represent the object as a point or a simple shape.

2. Identify all forces acting on the object. Consider all forces such as gravity, normal force, friction, applied forces, etc.

3. Ensure each force is exerted on the object of interest by another object. Forces should be identified by their source.

4. Draw vector arrows representing all forces acting on the object. The length of the arrows should be proportional to the magnitude of the forces, and the direction should be accurate.

2.33 Equilibrium

• For an object in equilibrium (constant velocity):

  • $\Sigma \vec{F} = 0$

  • $\Sigma F<em>x = 0$ and $\Sigma F</em>y = 0$ (and $\Sigma F_z = 0$). This means that the sum of the forces in each direction is zero.

2.34 Equilibrium

• Example 2.6: A hawk weighing 8 N is gliding north at constant speed. The total force due to the air must be 8 N upward to counter the gravitational force. Since the hawk is not accelerating, the net force on it must be zero.

2.35 Equilibrium

• Example 2.7: Forces on an airplane: gravity = 16.0 kN (downward), lift = 16.0 kN (upward), thrust = 1.8 kN (east), drag = 0.8 kN (west).

• $\Sigma F<em>x = L</em>x + T<em>x + W</em>x + D_x$ = 0 + 1.8 kN + 0 + (-0.8 kN) = 1.0 kN.

• $\Sigma F<em>y = L</em>y + T<em>y + W</em>y + D_y$ = (16 kN) + 0 + (-16 kN) + 0= 0.

• The net force is 1.0 kN east. This net force causes the airplane to accelerate in the eastward direction.

2.36 Equilibrium

• Example 2.8: Sliding a 750 N chest at constant velocity requires a 450 N horizontal force. Find the contact force that the floor exerts on the chest.

• $\Sigma F<em>x = W</em>x + F<em>x + C</em>x= 0 + 450 \text{ N} + C_x = 0$

• $\Sigma F<em>y = W</em>y + F<em>y + C</em>y= -750 \text{ N} + 0 + C_y = 0$

• Therefore, $C<em>x = -450 \text{ N}$ and $C</em>y = +750 \text{ N}$

• $C = \sqrt{(-450 \text{ N})^2 + (750 \text{ N})^2} = 870 \text{ N}$

• $\theta = \tan^{-1}{\frac{750 \text{ N}}{450 \text{ N}}} = 59^\circ$

2.37 Interaction Pairs: Newton’s Third Law

• Every force is part of an interaction between two objects; each exerts a force on the other. These forces form an interaction pair. These forces are equal in magnitude and opposite in direction.

2.38 Newton’s Third Law of Motion

• In an interaction between two objects, each exerts a force on the other.

• These forces are equal in magnitude and opposite in direction. This law is often stated as "For every action, there is an equal and opposite reaction."

2.39 Gravitational Forces

• Newton’s law of universal gravitation: Any two objects exert gravitational forces on each other proportional to their masses ($m<em>1$ and $m</em>2$) and inversely proportional to the square of the distance (r) between their centers. The formula for the gravitational force is $F = G \frac{m<em>1 m</em>2}{r^2}$, where G is the gravitational constant.

2.40 Mass Versus Weight

• Relationship between mass and weight: $\vec{W} = m\vec{g}$

• $\vec{W}$: Gravitational force (weight). Weight is a force and is measured in Newtons (N) or pounds (lb).

• $\vec{g}$: Gravitational field (direction is downward). The gravitational field is the force per unit mass exerted by a gravitational source.

• g (italicized): Magnitude of the gravitational field (never negative). The magnitude of the gravitational field near the Earth's surface is approximately 9.8 m/s^2.

• Average value of g near Earth’s surface: $9.80 \text{ N/kg}$

2.41 Weight

• Example 2.11: Weight of 350 g of fresh figs.

• Weight in newtons: $W = mg = 0.350 \text{ kg} \times 9.80 \text{ N/kg} = 3.4 \text{ N}$

• Weight in pounds: $3.4 \text{ N} \times 0.2248 \text{ lb/N} = 0.77 \text{ lb}$

2.42 Contact Forces

• Components of a contact force:

  • Normal force

  • Frictional force

• Normal Force: A contact force perpendicular to the contact surface. It prevents objects from passing through each other.

2.43 Friction

• Friction: A contact force parallel to the contact surface.

  • Static friction

  • Kinetic (or sliding) friction

2.44 Static Friction

• Maximum force of static friction:

• $f<em>{s,\text{max}} = \mu</em>s N$

• $\mu_s$: Coefficient of static friction (depends on the surfaces). It is a dimensionless quantity that represents the ratio of the maximum static friction force to the normal force.

2.45 Kinetic Friction

• Force of kinetic (sliding) friction: $f<em>k = \mu</em>k N$

• $\mu_k$: Coefficient of kinetic friction (depends on the surfaces). It is a dimensionless quantity that represents the ratio of the kinetic friction force to the normal force.

• \mus > \muk for an object on a given surface. This means that it takes more force to start an object moving than to keep it moving.

2.46 Friction

• Static friction acts to prevent objects from starting to slide; kinetic friction acts to stop sliding. Static friction opposes the initiation of motion, while kinetic friction opposes the continuation of motion.

• Frictional forces come in interaction pairs (Newton’s third law). When an object exerts a frictional force on a surface, the surface exerts an equal and opposite frictional force on the object.

2.47 Friction

• Example 2.12: Sliding a 750 N chest with a 450 N force.

• Coefficient of kinetic friction: $\mu<em>k = \frac{f</em>k}{N} = \frac{450 \text{ N}}{750 \text{ N}} = 0.60$