⚖️ Newton's Second Law

Understanding Force–Motion Relationships Through Problems

Newton's Second Law provides the fundamental relationship between the forces acting on a particle and its resulting acceleration.

ΣF = m a

The animations in this section use a series of representative problems to demonstrate how Newton's Second Law is formulated and applied to increasingly complex motions. Each problem connects the physical motion, free-body diagram, kinematic relations, and force balance.

🧮 Governing Principle

For a particle of mass m,

ΣF = m a

The vector equation can be resolved into appropriate coordinate directions depending on the geometry and motion of the problem.

The correct formulation requires both:

• a physically consistent free-body diagram (FBD), and

• the correct kinematic expression for acceleration.

🎥 Problems Explored

1. 🪀 Non-linear Pendulum Motion

Problem: A particle moves along a circular pendulum path under the action of gravity and the tension in the connecting member.

Concept: Application of Newton's Second Law to curvilinear motion using polar coordinates, with the acceleration resolved into radial and transverse components.

Watch For:

• How the gravitational force is resolved into the radial and transverse directions.

• How the tension contributes to the radial force balance.

• How the radial and transverse components of Newton's Second Law are satisfied independently at the scalar level.

• How the transverse acceleration, at = l θ̈ , governs the change in the particle's speed.

• How the radial acceleration, aᵣ = −l θ̇² , is associated with the curvature of the circular path. 

• Observe the live verification showing that the scalar force balance in each unit-vector direction matches the corresponding component of m a.

2. 🏹 Projectile Motion with Aerodynamic Drag

Problem: A projectile is launched into the air and experiences aerodynamic drag in addition to its weight.

Concept: Application of Newton's Second Law when the force acting on a particle depends on its instantaneous velocity, with the equations resolved along fixed Cartesian directions.

Watch For:

• How the aerodynamic drag force always acts opposite to the instantaneous velocity.

• How the changing velocity continuously changes both the magnitude and direction of the drag force.

• How the drag force introduces coupling between the horizontal and vertical components of motion.

• How Newton's Second Law is satisfied independently in the î and ĵ directions at the scalar level.

• How the trajectory, velocity, and acceleration differ from those of an ideal projectile without aerodynamic drag.

3. 🛝 Block Sliding on a Moving Frictionless Wedge

Problem: A block slides down a frictionless inclined wedge that is free to move horizontally on a smooth surface.

Concept: Application of Newton's Second Law to a system involving relative motion. The absolute acceleration of the block is first determined from the relative-motion kinematics and the motion of the wedge, before applying Newton's Second Law.

Watch For:

• The separate free-body diagrams for the block and the wedge.

• How the normal reaction acts on the block and wedge with equal magnitude and opposite directions.

• How the block's absolute acceleration is obtained by combining its relative acceleration with the acceleration of the wedge.

• How the kinematic relation between the relative and absolute motions couples the equations for the block and wedge.

• How Newton's Second Law is satisfied independently for each body.

• Observe the live verification showing ΣF = m a for both the block and the wedge.

4. 🌀 Pushing a Coin on a Rotating Disc

Problem: A coin is pushed radially inward on a rotating disc while the disc undergoes angular acceleration. Kinetic friction acts between the coin and the disc.

Concept: Application of Newton's Second Law to a particle undergoing relative motion in a rotating reference frame. The absolute acceleration of the coin is first determined from the rotating-frame kinematics before applying Newton's Second Law to the actual forces acting on the coin.

Watch For:

• How the absolute acceleration of the coin is composed of centripetal, tangential, and Coriolis components.

• How the centripetal acceleration arises from the rotation of the disc.

• How the tangential acceleration arises from the angular acceleration of the disc.

• How the Coriolis acceleration arises from the coin's radial motion relative to the rotating disc.

• How kinetic friction acts opposite to the relative motion of the coin.

• How the applied force changes as the angular velocity of the disc increases.

• How Newton's Second Law is satisfied component-by-component for the selected coordinate directions.

• Observe the three-dimensional free-body diagram and the live verification of ΣF = m a.

This is also a particularly nice place to reinforce the principle from the kinetics landing page:

⚠️ DO NOT USE PSEUDO-FORCES.

First determine the absolute (inertial) acceleration of the particle using the appropriate relative-motion kinematics. Then apply Newton's Second Law using only the actual forces acting on the particle.

5. 🔄 Collar Sliding on a Rough Horizontal Circular Track

Problem: A small collar of mass m is given an initial velocity v₀ on a horizontal circular track of radius r. The coefficient of kinetic friction is μₖ. Determine the distance travelled by the collar before it comes to rest.

Concept: Application of Newton's Second Law to circular motion using the tangent–normal coordinate system, with kinetic friction determined by the resultant normal force acting on the collar.

🔍 Watch For:

• How the collar moves along the circular path while its speed continuously decreases due to friction.

• How the tangent–normal unit vectors eₜ and eₙ rotate continuously with the collar.

• How the velocity v is always tangent to the path.

• How the acceleration is resolved into tangential and normal components, with the tangential component changing the speed and the normal component maintaining the circular path.

• How the radial normal force depends on the instantaneous speed of the collar.

• How the vertical support reaction balances the weight.

• How the kinetic friction force depends on the resultant normal force, and therefore changes as the collar slows down.

• How friction always acts opposite to the direction of motion.

• How the travelled distance s is tracked along the circular path until the collar comes to rest.

• Observe the 3-D free-body diagram showing the forces acting on the collar and their relationship with the tangent–normal directions.