🍏Principles of Physics I Unit 4 – Dynamics – Newton's Laws of Motion

Key Concepts

• Dynamics studies the forces that cause objects to move and how those forces affect the motion of objects
• Newton's laws of motion provide the foundation for understanding the relationship between forces and motion
• Forces are pushes or pulls that can cause an object to change its velocity (accelerate) or change its shape
• Free body diagrams visually represent all the forces acting on an object, allowing for a systematic analysis of the object's motion
• Friction is a force that opposes the relative motion between two surfaces in contact and can significantly affect an object's motion
• Circular motion requires a centripetal force, which is always directed toward the center of the circular path
• Problem-solving strategies, such as drawing diagrams, identifying known and unknown variables, and applying appropriate equations, are essential for successfully analyzing problems in dynamics

Newton's Laws Explained

• Newton's first law (law of inertia) states that an object at rest stays at rest, and an object in motion stays in motion with the same velocity, unless acted upon by an unbalanced external net force
  • Objects tend to resist changes in their state of motion, a property known as inertia
  • The mass of an object is a measure of its inertia; objects with greater mass require greater forces to change their motion
• Newton's second law relates the net external force acting on an object to its resulting acceleration: $\vec{F}_{net} = m\vec{a}$
  • The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass
  • The direction of the acceleration is always in the same direction as the net force
• Newton's third law states that when two objects interact, they apply forces to each other that are equal in magnitude and opposite in direction (action-reaction force pairs)
  • These action-reaction force pairs always act on different objects and cannot cancel each other out
  • Examples of action-reaction force pairs include the force of a person pushing a wall (person exerts force on wall, wall exerts equal and opposite force on person) and the force of Earth's gravity pulling on an object (Earth pulls on object, object pulls on Earth)

Forces and Free Body Diagrams

• Forces can be classified as contact forces (friction, tension, normal force) or non-contact forces (gravity, electrostatic force, magnetic force)
  • Contact forces require physical contact between the interacting objects, while non-contact forces can act over a distance
• The net force is the vector sum of all forces acting on an object and determines the object's acceleration according to Newton's second law
• Free body diagrams are used to identify and visualize all the forces acting on an object
  • Each force is represented by an arrow, with the arrow's length proportional to the force's magnitude and the arrow's direction indicating the force's direction
  • The object is typically represented as a dot or a simplified shape, with the forces acting on it drawn as arrows originating from the object
• When drawing free body diagrams, it is essential to include all relevant forces, such as gravity, friction, tension, and normal forces
  • The normal force is the force exerted by a surface on an object in contact with it, and it is always perpendicular to the surface
  • Tension forces occur in ropes, strings, or cables that are pulling on an object

Applying Newton's Laws

• To apply Newton's laws, first identify the object of interest (the system) and draw a free body diagram representing all the forces acting on it
• Determine the net force acting on the object by adding the force vectors, considering their magnitudes and directions
  • If the net force is zero, the object is in equilibrium and will either remain at rest or move with constant velocity
  • If the net force is non-zero, the object will accelerate in the direction of the net force
• Use Newton's second law ($\vec{F}_{net} = m\vec{a}$) to relate the net force to the object's acceleration
  • If the mass and net force are known, the acceleration can be calculated
  • If the mass and acceleration are known, the net force can be calculated
• When multiple objects are connected or interacting, it may be necessary to consider the forces acting on each object separately and then combine the equations to solve for the desired quantities
  • For example, in an Atwood machine (two masses connected by a pulley and rope), the tension in the rope can be found by applying Newton's second law to each mass and then solving the resulting system of equations

Motion in One and Two Dimensions

• Motion in one dimension (linear motion) occurs along a straight line and can be described using the equations of kinematics
  • These equations relate displacement ($\Delta x$), velocity ($v$), acceleration ($a$), and time ($t$): $v = v_0 + at$, $\Delta x = v_0t + \frac{1}{2}at^2$, $v^2 = v_0^2 + 2a\Delta x$
  • The choice of equation depends on the known and unknown variables in the problem
• Motion in two dimensions can be analyzed by treating the motion in each dimension (x and y) separately, using the one-dimensional equations of kinematics for each component
  • Projectile motion is an example of two-dimensional motion, where an object is launched with an initial velocity and follows a parabolic path under the influence of gravity
  • The horizontal and vertical components of a projectile's motion can be analyzed independently, with the horizontal velocity remaining constant (assuming no air resistance) and the vertical velocity changing due to the acceleration of gravity

Friction and Its Effects

• Friction is a force that opposes the relative motion between two surfaces in contact
  • Static friction acts between surfaces that are not moving relative to each other, and it can prevent an object from starting to move
  • Kinetic friction acts between surfaces that are moving relative to each other, and it always opposes the direction of motion
• The magnitude of the friction force depends on the normal force between the surfaces and the coefficient of friction ($\mu$), which is a property of the materials in contact
  • The static friction force can range from zero up to a maximum value given by $f_{s,max} = \mu_s F_N$, where $\mu_s$ is the coefficient of static friction and $F_N$ is the normal force
  • The kinetic friction force is given by $f_k = \mu_k F_N$, where $\mu_k$ is the coefficient of kinetic friction
• Friction can have a significant impact on an object's motion, causing it to slow down, come to rest, or require a greater force to maintain motion
  • When analyzing problems involving friction, it is essential to include the friction force in the free body diagram and to consider its direction and magnitude

Circular Motion and Centripetal Force

• Circular motion is a type of motion in which an object follows a circular path with a constant speed
  • The velocity vector is always tangent to the circular path and perpendicular to the radius
  • Even though the speed is constant, the object experiences acceleration because the direction of the velocity vector is constantly changing
• The acceleration in circular motion is called centripetal acceleration and is always directed toward the center of the circular path
  • The magnitude of the centripetal acceleration is given by $a_c = \frac{v^2}{r}$, where $v$ is the object's speed and $r$ is the radius of the circular path
• For an object to undergo circular motion, there must be a net force acting on it that is always directed toward the center of the circular path, known as the centripetal force
  • The magnitude of the centripetal force is given by $F_c = m\frac{v^2}{r}$, where $m$ is the object's mass
  • Various forces can provide the necessary centripetal force, such as gravity (planets orbiting the sun), tension (a mass attached to a string and swung in a circle), or friction (a car making a turn)

Problem-Solving Strategies

• Read the problem carefully and identify the given information, the unknown quantities, and the relevant concepts or principles
• Draw a diagram (such as a free body diagram) to visually represent the problem and the forces acting on the object(s) of interest
• Identify the coordinate system and the positive direction for each axis
• Break the problem down into smaller, more manageable steps, and solve for the unknown quantities using the appropriate equations
  • If the problem involves multiple objects or dimensions, consider each object or dimension separately and then combine the results as needed
• Check the units of your answer to ensure they are consistent with the quantity you are solving for
• Evaluate the reasonableness of your answer based on the problem's context and your understanding of the physical principles involved
  • If the answer seems unreasonable, review your solution process to identify any errors or misunderstandings
• Practice solving a variety of problems to develop your problem-solving skills and deepen your understanding of the concepts in dynamics