Applications of Newton's Laws
Applications of Newton's Laws
Chapter Objectives
By the end of this chapter, you will be able to:
Draw a free-body diagram showing all forces acting on an individual object, representing them as vectors.
Solve for unknown quantities (such as magnitudes of forces or accelerations) using Newton’s second law in problems involving an individual object or a system of objects connected to each other, including pulleys and inclined planes.
Relate the force of friction acting on an object to the normal force exerted on an object using appropriate coefficients of friction in Newton’s second law problems.
Use Hooke’s law to relate the magnitude of the spring force exerted by a spring to the distance from the equilibrium position the spring has been stretched or compressed (F_{spring} = -kx).
Overview of Newton’s Laws
Newton’s three laws of motion are foundational principles of classical mechanics, describing the relationship between an object's motion and the forces acting upon it. Effectively applying these laws requires specific problem-solving skills.
This chapter focuses on developing advanced problem-solving techniques, beginning with equilibrium problems (where acceleration is zero) and then advancing to analyzing dynamic situations involving net forces and motion.
Equilibrium of a Particle
An object is in equilibrium when it is either at rest (static equilibrium) or moving with a constant velocity (dynamic equilibrium) in an inertial frame of reference.
An inertial frame is a non-accelerating reference frame where Newton's first law holds true.
Examples of objects in equilibrium include:
A hanging lamp: The tension in the chain balances the gravitational force on the lamp.
A rope-and-pulley system for hoisting loads: When lifting at a constant speed, the net force on the load and moving parts is zero.
A suspension bridge: The forces in the cables and structural components are balanced, supporting the bridge deck against gravity.
Equilibrium Conditions
Newton’s First Law of Motion: An object at rest stays at rest, and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force. Consequently, the vector sum of all external forces acting on an object in equilibrium must be zero.
This can be expressed mathematically as:
\sum \vec{F} = 0
In component form, for an object in equilibrium, the sum of the force components along each axis must be zero:
\sum F_x = 0
\sum F_y = 0
(For three-dimensional problems, \sum F_z = 0 would also apply).
Choice of Coordinate System
You may choose any convenient coordinate system (e.g., Cartesian x-y axes) to resolve forces into components. The choice should simplify the problem, often by aligning one axis with a direction where many forces act or where potential motion might occur.
Components may be positive or negative depending on the chosen positive directions for each axis.
Necessary Condition for Equilibrium: A net force of zero (\sum \vec{F} = 0) implies that no acceleration is occurring.
An object in equilibrium could therefore be at rest or moving with constant velocity.
Important equations derived from the component form of Newton's First Law:
\sum F_x = 0
\sum F_y = 0
Problem-Solving Strategy for Equilibrium Problems
Set Up
Draw a detailed sketch: Illustrate the physical apparatus or structure, showing all relevant dimensions, angles, and any objects involved. Label known quantities clearly.
Identify the object(s) in equilibrium: Clearly define the system or particle you are analyzing. This is often an object at rest or moving at constant velocity.
Draw a free-body diagram (FBD) for each identified object: Isolate the object and draw only the external forces acting on it. Represent each force as a vector originating from a single point on the object, indicating its correct direction and label (e.g., Weight (W or mg), Normal force (N), Tension (T), Spring force (F_{sp}), Friction (f)). Do not include forces the object exerts on other objects, and do not include velocity or acceleration vectors on the FBD itself.
Choose coordinate axes and resolve forces into components: Select an orthogonal coordinate system that simplifies the resolution of forces. For forces not aligned with the axes, break them down into their x and y components using trigonometry (e.g., Fx = F \cos\theta, Fy = F \sin\theta).
Apply equilibrium conditions: Set the algebraic sum of the force components in the x-direction and y-direction equal to zero (i.e., \sum Fx = 0 and \sum Fy = 0). This will yield a system of simultaneous equations.
Solve for unknowns: Solve the system of equations for the unknown quantities (e.g., magnitudes of forces, angles).
For systems of multiple objects: If multiple interconnected objects are in equilibrium, draw a separate FBD for each object and use Newton’s third law (action-reaction pairs between objects) to relate forces between them.
Reflect on results for reasonability: Check if the calculated magnitudes, directions, and units make logical sense in the physical context of the problem. For example, tensions should be positive, and forces should not be excessively large or small without reason.
Example Problem: Tension in a Rope
Consider a gymnast hanging from a rope connected to an O-ring bolted to the ceiling. The weights are given as 500 N (gymnast), 100 N (rope), and 50 N (O-ring).
To find the tension supporting the gymnast, we isolate the gymnast as the object in equilibrium. The forces acting on the gymnast are the downward gravitational force (weight) and the upward tension (T) from the rope.
Applying the equilibrium condition in the vertical (y) direction:
\sum F_y = T - 500 N = 0 \Rightarrow T = 500 N
This means the tension in the rope segment directly supporting the gymnast is 500 N.
Two-Dimensional Equilibrium Example
A common scenario involves a car engine (weight w) suspended by two chains that are at different angles to the horizontal or vertical. This requires careful resolution of forces.
The problem involves drawing a free-body diagram for the point where the chains meet (often treated as a particle) and using trigonometry to break down the tension forces in the chains into their horizontal (x) and vertical (y) components. For instance: when an angle \theta is given with respect to the horizontal, the horizontal component is F \cos\theta and the vertical component is F \sin\theta.
The equations will then sum forces in both the x and y directions to zero: \sum Fx = 0 and \sum Fy = 0. These simultaneous equations can then be solved for unknown tension quantities, even if example geometry includes varying angles such as 60^\circ or 45^\circ.
Applications of Newton’s Second Law
According to Newton's Second Law: 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 in the direction of the net force.
Statement of Newton’s Second Law
In its full vector form, Newton's Second Law is stated as:
\sum \vec{F} = m\vec{a}
In component form, for problems typically in two or three dimensions, this equation can be broken down into scalar equations along each axis:
\sum Fx = m ax
\sum Fy = m ay
\sum Fz = m az
Here, ax, ay, and a_z are the components of the acceleration vector \vec{a}.
Problem-Solving Strategy for Non-Equilibrium Situations
Draw a sketch: Create a visual representation of the moving objects and their environment. Include initial/final positions, velocities, and the expected direction of acceleration if known.
Draw a free-body diagram (FBD) for each object: Isolate each object and show all external forces acting on it. Represent forces as vectors, and add an arrow indicating the direction of the object's acceleration alongside the FBD (but not on the FBD itself) to guide axis selection and equation setup.
Choose coordinate axes and align with acceleration: Crucially, identify the direction of acceleration for each object. Select your coordinate axes such that one axis (e.g., the x-axis) is aligned with the direction of acceleration. This often simplifies the problem by making one component of acceleration zero (a_y = 0 if x is the direction of motion).
Resolve the forces into components: Break down any forces that are not parallel to your chosen coordinate axes into their x and y components using trigonometry.
Apply Newton’s second law: Write down the equations for Newton's second law for each axis for each object (\sum Fx = m ax, \sum Fy = m ay). Solve the resulting system of simultaneous equations for the unknown quantities (e.g., accelerations, tensions, or specific forces).
Reflect on results for reasonability: Evaluate your answers. Do the magnitudes and directions of forces and accelerations make physical sense? Are the units correct? For example, an object accelerating down an incline should have an acceleration less than g.
Example Problem: Hockey Puck
Consider a hockey puck sliding across an ice surface after being hit. The primary horizontal force acting against its motion is kinetic friction.
By drawing a free-body diagram and applying Newton's second law, the force of kinetic friction can be used to compute the puck's deceleration (negative acceleration). This average acceleration, obtained from the forces acting on the system, can then be used in kinematic equations to determine how long the puck slides or how far it travels before coming to rest.
Contact Forces and Friction
The definitions of normal force and friction forces are critical for analyzing the motion of objects in contact with surfaces.
Normal Force (N or n): This is a contact force exerted by a surface on an object that is in contact with it. It always acts perpendicular to the surface of contact, pushing outwards from the surface. The normal force prevents objects from passing through surfaces and is not always equal to the object's weight, especially on inclined planes or when other vertical forces are present.
Friction Force (f): This is a contact force that acts parallel to the surface of contact, opposing the relative motion or the tendency of relative motion between the surfaces.
Kinetic Friction (f_k):
Occurs when two surfaces are in motion relative to each other (e.g., sliding).
Its magnitude is generally constant for a given pair of surfaces and is proportional to the normal force.
The relationship is given by: fk = \muk N
Here, \muk (mu-k) is the coefficient of kinetic friction, a dimensionless constant that depends on the properties of the two surfaces in contact. \muk is typically less than \mu_s.
Static Friction (f_s):
Occurs when two surfaces are at rest relative to each other, but there is an external force attempting to cause relative motion.
Cheat Sheet Formulas
Hooke's Law for spring force:
F_{\text{spring}} = -kx
Used to calculate the force exerted by a spring, where k is the spring constant and x is the displacement from equilibrium.
Newton's First Law (Equilibrium Condition):
\sum \vec{F} = 0
States that the vector sum of all external forces acting on an object in equilibrium (at rest or constant velocity) must be zero.
Newton's First Law in component form (Equilibrium Conditions):
Newton's First Law in component form (Equilibrium Conditions):
\sum Fx = 0
\sum Fy = 0
Used to analyze equilibrium problems by setting the sum of force components along each axis to zero.
\sum F_z = 0
Used to analyze equilibrium problems by setting the sum of force components along each axis to zero.
Force components using trigonometry:
Force components using trigonometry:Fx = F \cos\theta Fy = F \sin\thetaUsed to resolve a force vector F into its horizontal (x) and vertical (y) components when the angle \theta with respect to the horizontal is known.
Used to resolve a force vector F into its horizontal (x) and vertical (y) components when the angle \theta with respect to the horizontal is known.
Newton's Second Law (Vector Form):
\sum \vec{F} = m\vec{a}
Relates the net force acting on an object to its mass (m) and acceleration (\vec{a}).
Newton's Second Law in component form:
\sum Fx = m ax \sum Fy = m ay \sum Fz = m az Used to solve non-equilibrium problems by relating the sum of force components along each axis to the respective components of acceleration.
Used to solve non-equilibrium problems by relating the sum of force components along each axis to the respective components of acceleration.
Kinetic Friction Force:
𝑓𝑘=𝜇𝑘𝑁