Applications of Newton's Laws by Owen Borville 11.21.2025
Friction is a contact force that opposes motion between two surfaces in contact and acting in direction opposite to the movement.
Simple friction is proportional to the normal force N pushing the systems together. A normal force is always perpendicular to the contact surface between systems or objects in contact. Friction depends on both of the materials involved.
Normal force is not always equal in magnitude to the weight of the object. If an object is accelerating vertically, the normal force is less than or greater than the weight of the object. Also, if the object is on an inclined plane, the normal force is always less than the full weight of the object.
Direction of acceleration of an object determines whether Fnet = ma or Fnet = 0.
The magnitude of static friction between systems stationary relative to one another is fs <= μsN, where μs is the coefficient of static friction, which depends on both of the materials.
The kinetic friction force fk between systems moving relative to one another is: fk = μkN, where μk is the coefficient of kinetic friction, also dependent of both materials and is always less than the coefficient of static friction.
Centripetal force Fc (vector) is a center-seeking force that always points toward the center of rotation motion. It is perpendicular to linear velocity and has the magnitude Fc = mac or Fc = m(v^2/r) or Fc = m(r𝜔^2)
Rotating and accelerated frames of reference are non-inertial. Inertial forces, such as the Coriolis force are needed to explain motion in such frames.
Drag forces acting on an object moving in a fluid oppose the motion: Fd = 1/2C𝜌Av^2, where C is the drag coefficient, A is the area of the object facing the fluid, and 𝜌 is the fluid density.
For small objects like bacterium, moving in a denser medium such as water, the drag force is Stokes Law: Fs = 6 π r 𝜂 v, where r is the radius of the object, 𝜂 is the fluid viscosity, and v is the object's velocity.
The ideal angle of a banked curve is = tan θ = v^2/rg
Elasticity: Stress and Strain: Hooke's law=F=k ΔL, where ΔL is the amount of deformation or change in length, F is the applied force, and k is a proportionality constant that depends on the shape and composition of the object and the direction of the force.
Deformation and applied force can be related by ΔL = 1/Y(F/A)L0, where Y is young's modulus, a value that depends on the substance, A is the cross-sectional area, and L0 is the original length. The ratio of force to area, F/A, is defined as stress, and measured in N/m^2.
The ratio of change in length to length, ΔL/L0, is defined as strain (a unitless quantity). Stress = Y*(strain)
Shear deformation = Δx = 1/S(F/A)L0, where S is the shear modulus and F is the force applied perpendicular to L0 and parallel to the cross-sectional area A.
The relationship of the change in volume to other physical quantities is = ΔV = 1/B(F/A)V0, where B is the bulk modulus, V0 is the original volume, and F/A is the force per unit area applied uniformly inward on all surfaces.
Rotation Angle and Angular Velocity: Uniform circular motion is motion in a circle at constant speed. The rotation angle Δθ is defined as the ratio of the arc length to the radius of curvature:
Δθ = Δs/r,
where arc length Δs is the distance travelled along a circular path and r is the radius of curvature of the circular path. The quantity Δθ is measured in units of radians (rad):
2π rad = 360° = 1 revolution.
The conversion between radians an degrees is 1 rad = 57.3°.
Angular velocity (ω) (lower omega) is the rate of change of an angle: ω = Δθ/ Δt, where a rotation Δθ takes place during a time Δt. The units of angular velocity are radians per second (rad/sec). Linear velocity v and angular velocity ω are related by v = r ω or ω = v/r
Centripetal Acceleration (ac) is the acceleration experienced while in uniform circular motion. Ac always points toward the center of rotation and is perpendicular to the linear velocity v and has the magnitude ac = v^2/r and ac = rω^2. The unit of centripetal acceleration is m/s^2.
Centripetal Force (Fc) is any force that causes uniform circular motion. (Fc) is a center-seeking force that always points toward the center of rotation. (Fc) is perpendicular to linear velocity v and has magnitude Fc = ma(c), or
Fc = mv^2/r or
Fc = mrω ^2
Fictitious forces and non-inertial frames include the Coriolis Force. Rotating and accelerated frames of reference are non-inertial. Fictitious forces, such as the Coriolis force, are needed to explain motion in such frames. The Coriolis force is an apparent force that causes a deflection in the path of moving objects, such as wind and ocean currents, due to the rotation of the Earth. It is not a real force like gravity but it is an effect observed within a rotating frame of reference. This effect causes winds to curve to the right in the Northern hemisphere and to the left in the Southern hemisphere, leading to the rotation of hurricanes and influencing weather patterns.
Newton's Universal Law of Gravitation: Every particle in the universe attracts every other particle with a force along a line joining them. The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them: F = G(mM/r^2), where F is the magnitude of the gravitational force. G is the gravitational constant, given by G = 6.674 x 10^-11 N*m^2/kg^2. Newton's law of gravitation applies universally.
Kepler's Laws of Planetary Satellites (Johannes Kepler, 1571-1630) Kepler's Laws are stated for a small mass m orbiting a larger mass M in near-isolation. Kepler's laws of planetary motion are:
Kepler's 1st Law: The orbit of each planet about the Sun is an ellipse with the Sun at one focus.
Kepler's 2nd Law: Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal times.
Kepler's 3rd Law: The ratio of the squares of the periods of any two planets about the Sun is equal to the ratio of the cubes of their average distances from the Sun: (T1^2/T2^2)=(r1^3/r2^3), where T is the period (time for one orbit) and r is the average radius of the orbit.
The period and radius of a satellite's orbit about a larger body M are related by: T^2 = (4 π^2/GM)r^3 or r^3/T^2 = (G/4 π^2)M
Friction is a contact force that opposes motion between two surfaces in contact and acting in direction opposite to the movement.
Simple friction is proportional to the normal force N pushing the systems together. A normal force is always perpendicular to the contact surface between systems or objects in contact. Friction depends on both of the materials involved.
Normal force is not always equal in magnitude to the weight of the object. If an object is accelerating vertically, the normal force is less than or greater than the weight of the object. Also, if the object is on an inclined plane, the normal force is always less than the full weight of the object.
Direction of acceleration of an object determines whether Fnet = ma or Fnet = 0.
The magnitude of static friction between systems stationary relative to one another is fs <= μsN, where μs is the coefficient of static friction, which depends on both of the materials.
The kinetic friction force fk between systems moving relative to one another is: fk = μkN, where μk is the coefficient of kinetic friction, also dependent of both materials and is always less than the coefficient of static friction.
Centripetal force Fc (vector) is a center-seeking force that always points toward the center of rotation motion. It is perpendicular to linear velocity and has the magnitude Fc = mac or Fc = m(v^2/r) or Fc = m(r𝜔^2)
Rotating and accelerated frames of reference are non-inertial. Inertial forces, such as the Coriolis force are needed to explain motion in such frames.
Drag forces acting on an object moving in a fluid oppose the motion: Fd = 1/2C𝜌Av^2, where C is the drag coefficient, A is the area of the object facing the fluid, and 𝜌 is the fluid density.
For small objects like bacterium, moving in a denser medium such as water, the drag force is Stokes Law: Fs = 6 π r 𝜂 v, where r is the radius of the object, 𝜂 is the fluid viscosity, and v is the object's velocity.
The ideal angle of a banked curve is = tan θ = v^2/rg
Elasticity: Stress and Strain: Hooke's law=F=k ΔL, where ΔL is the amount of deformation or change in length, F is the applied force, and k is a proportionality constant that depends on the shape and composition of the object and the direction of the force.
Deformation and applied force can be related by ΔL = 1/Y(F/A)L0, where Y is young's modulus, a value that depends on the substance, A is the cross-sectional area, and L0 is the original length. The ratio of force to area, F/A, is defined as stress, and measured in N/m^2.
The ratio of change in length to length, ΔL/L0, is defined as strain (a unitless quantity). Stress = Y*(strain)
Shear deformation = Δx = 1/S(F/A)L0, where S is the shear modulus and F is the force applied perpendicular to L0 and parallel to the cross-sectional area A.
The relationship of the change in volume to other physical quantities is = ΔV = 1/B(F/A)V0, where B is the bulk modulus, V0 is the original volume, and F/A is the force per unit area applied uniformly inward on all surfaces.
Rotation Angle and Angular Velocity: Uniform circular motion is motion in a circle at constant speed. The rotation angle Δθ is defined as the ratio of the arc length to the radius of curvature:
Δθ = Δs/r,
where arc length Δs is the distance travelled along a circular path and r is the radius of curvature of the circular path. The quantity Δθ is measured in units of radians (rad):
2π rad = 360° = 1 revolution.
The conversion between radians an degrees is 1 rad = 57.3°.
Angular velocity (ω) (lower omega) is the rate of change of an angle: ω = Δθ/ Δt, where a rotation Δθ takes place during a time Δt. The units of angular velocity are radians per second (rad/sec). Linear velocity v and angular velocity ω are related by v = r ω or ω = v/r
Centripetal Acceleration (ac) is the acceleration experienced while in uniform circular motion. Ac always points toward the center of rotation and is perpendicular to the linear velocity v and has the magnitude ac = v^2/r and ac = rω^2. The unit of centripetal acceleration is m/s^2.
Centripetal Force (Fc) is any force that causes uniform circular motion. (Fc) is a center-seeking force that always points toward the center of rotation. (Fc) is perpendicular to linear velocity v and has magnitude Fc = ma(c), or
Fc = mv^2/r or
Fc = mrω ^2
Fictitious forces and non-inertial frames include the Coriolis Force. Rotating and accelerated frames of reference are non-inertial. Fictitious forces, such as the Coriolis force, are needed to explain motion in such frames. The Coriolis force is an apparent force that causes a deflection in the path of moving objects, such as wind and ocean currents, due to the rotation of the Earth. It is not a real force like gravity but it is an effect observed within a rotating frame of reference. This effect causes winds to curve to the right in the Northern hemisphere and to the left in the Southern hemisphere, leading to the rotation of hurricanes and influencing weather patterns.
Newton's Universal Law of Gravitation: Every particle in the universe attracts every other particle with a force along a line joining them. The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them: F = G(mM/r^2), where F is the magnitude of the gravitational force. G is the gravitational constant, given by G = 6.674 x 10^-11 N*m^2/kg^2. Newton's law of gravitation applies universally.
Kepler's Laws of Planetary Satellites (Johannes Kepler, 1571-1630) Kepler's Laws are stated for a small mass m orbiting a larger mass M in near-isolation. Kepler's laws of planetary motion are:
Kepler's 1st Law: The orbit of each planet about the Sun is an ellipse with the Sun at one focus.
Kepler's 2nd Law: Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal times.
Kepler's 3rd Law: The ratio of the squares of the periods of any two planets about the Sun is equal to the ratio of the cubes of their average distances from the Sun: (T1^2/T2^2)=(r1^3/r2^3), where T is the period (time for one orbit) and r is the average radius of the orbit.
The period and radius of a satellite's orbit about a larger body M are related by: T^2 = (4 π^2/GM)r^3 or r^3/T^2 = (G/4 π^2)M