Kinetics of Particles: Force and Acceleration

Introduction

Kinetics relates the motion of a body to the forces and moments that produce it. In this chapter the body is idealized as a particle: its size and rotational motion are neglected, while its mass and translational motion are retained. This model is appropriate when the dimensions of a body are small compared with the distances involved, or when only the motion of its center of mass is required.

The central principle is Newton’s second law. For a particle of constant mass, the resultant force equals mass multiplied by acceleration. The law provides a systematic method for analyzing vehicles, elevators, projectiles, machine components, cable systems, and many other engineering systems.

Learning Objectives

After studying this chapter, the student should be able to:

1. Draw an appropriate free-body diagram for a particle.
2. Apply Newton’s second law in rectangular, normal–tangential, and polar coordinates.
3. Relate force, acceleration, velocity, and displacement.
4. Analyze particles connected by inextensible cables or constrained to move along paths.
5. Model elementary drag and friction effects.
6. Interpret constraint forces and identify the assumptions behind particle models.

1. Newton’s Laws and the Particle Model

Newton’s first law states that a particle remains at rest or moves with constant velocity when the resultant external force is zero. Newton’s second law, in vector form, is

where ∑F is the resultant external force in newtons (N), m is mass in kilograms (kg), and a is acceleration in metres per second squared (m/s²). The equation is valid in an inertial reference frame and for constant mass.

Newton’s third law states that forces between two interacting bodies occur in equal and opposite pairs. These forces act on different bodies and therefore must not be cancelled on a single free-body diagram.

For variable-mass systems, such as rockets or leaking containers, the constant-mass form requires modification. Such systems are outside the basic particle model unless mass-flow effects are explicitly included.

Units and dimensions

Force has the SI unit newton:

Dimensional consistency is a useful check. Every term in a force equation must have dimensions of force. Weight is

where g ≈ 9.81 m/s² near Earth’s surface. Mass is measured in kilograms; weight is a force measured in newtons.

2. Free-Body Diagrams and Equations of Motion

A free-body diagram (FBD) isolates the particle and shows every external force acting on it. Common forces include:

• weight, directed vertically downward;
• normal force, perpendicular to a contact surface;
• tension, directed along a cable or rope;
• friction, tangent to a contact surface and opposing relative or impending motion;
• spring force, directed toward the spring’s undeformed position;
• applied force and fluid drag.

The recommended procedure is:

1. Define the particle or system being isolated.
2. Select positive coordinate directions.
3. Draw all known and unknown external forces.
4. Resolve forces into the selected directions.
5. Write one equation of motion for each independent direction.
6. Apply kinematic or geometric constraints.
7. Check signs, units, and physical plausibility.

For rectangular coordinates,

These equations are independent scalar components of the vector law. They apply when the axes are fixed in an inertial frame.

Example 1: Block pulled along a rough horizontal surface

A 20 kg block is pulled horizontally by a 100 N force. The coefficient of kinetic friction is 0.20. Determine its acceleration, assuming the surface is horizontal and the pulling force has no vertical component.

The vertical equation gives

Thus N = 20(9.81) = 196.2 N. Kinetic friction is

In the horizontal direction,

Therefore,

The acceleration is in the direction of the applied force. The calculation assumes kinetic sliding has already begun and that the coefficient of friction is constant.

3. Rectilinear Motion

When a particle moves along a straight line, its position, velocity, and acceleration may be described by one coordinate x. The definitions are

and

If acceleration is constant, integration gives

where x₀ and v₀ are the initial position and velocity. These constant-acceleration equations must not be used when acceleration varies with time, position, or velocity unless the variation has first been accounted for.

For variable acceleration, Newton’s law can be combined with integration:

The relation follows from the chain rule and is useful when force depends on position. If force depends on velocity, the differential equation is commonly written as

and solved by integration or numerical methods.

4. Curvilinear Motion

For motion along a curved path, acceleration has components that describe changes in speed and direction. Two particularly useful coordinate systems are normal–tangential and polar coordinates.

4.1 Normal–tangential coordinates

Let t be tangent to the path in the direction of increasing motion, and n point toward the instantaneous center of curvature. Then

with

and

where v is speed and ρ is the radius of curvature. Therefore Newton’s law becomes

The normal component exists even at constant speed because the velocity direction changes. If the path is straight, ρ → ∞ and a_n = 0.

Example 2: Vehicle on a banked curve

A 1200 kg vehicle travels at 15 m/s around a level circular curve of radius 50 m. Determine the required inward resultant force.

The normal acceleration is

Thus

The road, through friction and possibly banking, must supply a resultant horizontal inward force of 5.40 kN. This result does not by itself determine the normal force or friction force; those depend on the road geometry and contact conditions.

4.2 Polar coordinates

Polar coordinates describe position using radial distance r and angular position θ. The acceleration is

where dots denote time derivatives. The radial component is

and the transverse component is

The terms involving θ̇ represent effects of changing direction, while the term 2ṙθ̇ is sometimes called a transverse or Coriolis-type term in polar kinematics. Polar equations are especially useful for sliders in rotating arms and particles moving in spiral paths.

5. Circular Motion and Constraint Forces

For uniform circular motion, speed is constant, so a_t = 0. The acceleration is purely inward:

where r is the circle radius and ω is angular speed in rad/s. Since v = rω, the two forms are equivalent.

The inward force is not a new physical force called “centripetal force.” Rather, centripetal force is the name given to the resultant inward component of actual forces such as tension, friction, gravity, or a normal reaction.

A constraint force enforces permissible motion. For example, a cable tension may maintain a fixed distance, while a guide reaction may keep a slider on a track. Constraint forces do no work when they are always perpendicular to displacement, but this is not universally true; a constraint force can perform work if the constraint or contact moves.

6. Connected Particles and Cable Constraints

For ideal cables, the cable is assumed massless and inextensible, and pulleys are assumed smooth and frictionless. Under these assumptions, tension is uniform throughout one continuous cable. If the cable length is constant, the sum of its changing segments is constant.

For a simple two-block system connected by a cable over a fixed pulley, the magnitudes of acceleration are equal:

Their directions are opposite when measured using parallel upward-positive coordinates. The equations of motion must nevertheless be written separately for each block.

Example 3: Two connected blocks

A 6 kg block on a smooth horizontal table is connected over a frictionless pulley to a hanging 4 kg block. Determine the acceleration and cable tension.

For the 6 kg block,

For the hanging block, taking downward as positive,

Substitution gives

Therefore,

and

The tension is less than the hanging block’s weight, as expected, because the block accelerates downward.

For more complex cable arrangements, differentiate the cable-length equation once to obtain a velocity relation and twice to obtain an acceleration relation. Signs must be retained until the physical directions are interpreted.

7. Friction, Springs, and Drag

7.1 Friction

For impending or static contact,

The static friction force is whatever value is required for equilibrium or compatible motion, up to its limiting value μ_sN. It is incorrect to set F_s = μ_sN in every static situation.

For sliding contact,

where μ_s and μ_k are dimensionless coefficients. Usually μ_s ≥ μ_k, although actual values depend on materials, surface condition, temperature, lubrication, and speed.

7.2 Linear springs

For an ideal linear spring,

where k is stiffness in N/m and δ is deformation from the undeformed length. The spring force acts opposite the deformation. This relation is valid within the elastic linear range of the spring.

7.3 Fluid drag

A common model for high-speed motion through a fluid is quadratic drag:

where ρ is fluid density in kg/m³, C_D is a dimensionless drag coefficient, A is reference area in m², and v is speed relative to the fluid in m/s. Drag acts opposite relative motion. The coefficient depends on shape and Reynolds number, so it is not generally a universal constant.

At low Reynolds number, a linear model may be appropriate:

where c is a viscous damping coefficient in N·s/m. Model selection must be based on the flow regime and experimental or validated empirical data.

8. Work–Energy as an Alternative to Direct Time Integration

Although Newton’s second law is the primary focus, the work–energy relation can simplify problems in which force and displacement are known but time is not required. For a particle,

where kinetic energy is

and U₁→₂ is the work done by all forces from position 1 to position 2. For a variable force,

The method is valid for a particle in an inertial frame, with work evaluated consistently. It does not directly provide time unless an additional relation is used.

9. Detailed Application: Emergency Braking of an Autonomous Delivery Vehicle

Background

Consider an illustrative low-speed autonomous delivery vehicle of mass 800 kg moving on a level dry road at 12 m/s. A forward sensor detects an obstacle. The control system commands a braking force modelled as 5000 N opposite the motion. Rolling resistance is represented by 400 N opposite the motion.

Problem

Determine the deceleration and stopping distance, assuming both forces remain constant and tire adhesion is sufficient.

Analysis and solution

The total retarding force is

Taking the direction of motion as positive,

Hence

The stopping distance follows from v² = v₀² + 2aΔx with final speed zero:

Therefore,

The stopping time is

Outcome and engineering interpretation

The idealized vehicle stops in approximately 10.7 m after 1.78 s. A real safety analysis must add sensor-processing delay, actuator delay, road slope, tire-force limits, load variation, brake-temperature effects, and uncertainty in the drag and rolling-resistance models. If a 0.5 s control delay occurs before braking begins, the vehicle travels an additional 6.0 m, giving a total distance of approximately 16.7 m.

Learning points

1. A free-body diagram identifies all retarding forces.
2. Constant-force assumptions permit constant-acceleration equations.
3. System delay can be as important as mechanical braking distance.
4. Safety decisions require margins rather than reliance on ideal calculations.
5. Sensor and control software should be validated against physical limits and adverse conditions.

10. Common Errors and Best Practices

A frequent error is confusing mass and weight. Another is drawing action–reaction pairs on the same FBD and cancelling them. Students also often use v²/r without identifying the inward direction, or assume that constant speed means zero acceleration. In circular motion, acceleration is zero only when both speed and direction are constant; a curved path necessarily changes direction.

Best practice includes:

• isolate one body or a clearly defined system;
• choose axes that simplify the geometry;
• label positive directions before writing equations;
• distinguish known forces from reactions and constraint forces;
• use SI units consistently;
• retain signs until the result is interpreted;
• check limiting cases, such as zero friction or zero curvature;
• compare the result with physical expectations;
• state assumptions clearly in engineering reports.

Computational tools can integrate nonlinear equations involving drag or changing constraints, but software does not replace the FBD or model validation. Numerical results should be checked using units, conservation principles, limiting cases, and sensitivity to uncertain parameters.

Chapter Summary

Kinetics of particles is founded on Newton’s second law, ∑F = ma, applied to a carefully isolated particle in an inertial reference frame. Free-body diagrams provide the physical basis for the equations. Rectangular coordinates are effective for straight-line and Cartesian motion; normal–tangential coordinates separate changes in speed from changes in direction; and polar coordinates describe radial and angular motion.

For connected particles, cable-length constraints supplement Newton’s equations. Friction, spring forces, and fluid drag are models whose validity depends on contact, material, and flow assumptions. Circular motion requires an inward resultant force even when speed is constant. Work–energy offers an alternative when displacement and speed are more important than time. Reliable engineering analysis depends on correct force identification, unit consistency, explicit assumptions, and verification against physical behavior.

Key Terms

Particle: Idealized body whose size and rotation are neglected.

Kinetics: Study of motion in relation to applied forces.

Inertial frame: Reference frame in which Newton’s laws have their standard form.

Resultant force: Vector sum of all external forces acting on a body.

Free-body diagram: Diagram showing a selected body and all external forces on it.

Mass: Measure of resistance to acceleration, expressed in kilograms.

Weight: Gravitational force, W = mg.

Normal force: Contact force perpendicular to a surface.

Tension: Pulling force transmitted by a cable or rope.

Static friction: Friction force before sliding, limited by μ_sN.

Kinetic friction: Friction force during sliding, commonly μ_kN.

Constraint force: Reaction that enforces a geometrical or contact restriction.

Tangential acceleration: Component dv/dt along a path.

Normal acceleration: Component v²/ρ toward the center of curvature.

Radius of curvature: Local radius of a curved path.

Drag: Fluid resistance opposing relative motion.

Cable constraint: Geometrical relation imposed by an inextensible cable.

Centripetal resultant: Inward resultant force required for curved motion.

Linear spring: Ideal spring whose force is proportional to deformation.

Work–energy relation: Relation connecting work and change in kinetic energy.

Review Questions

Short Answer Questions

1. State Newton’s second law for a constant-mass particle and define each quantity.
2. Why must action–reaction forces not be cancelled on one particle’s free-body diagram?
3. What is the physical meaning of normal acceleration in circular motion?
4. State the difference between static and kinetic friction.
5. Identify two assumptions commonly made for an ideal cable and pulley.

Long Answer Questions

1. Explain a systematic procedure for constructing a free-body diagram and writing particle equations of motion.
2. Derive the normal–tangential acceleration components and discuss their use in vehicle-motion analysis.
3. Explain polar-coordinate acceleration and the physical significance of each term.
4. Compare linear and quadratic drag models, including the conditions under which each may be appropriate.
5. Discuss the advantages and limitations of using the work–energy method instead of direct application of Newton’s second law.

Analytical / Application Questions

1. A 10 kg package is pulled along a horizontal surface by a 75 N force at 30° above the horizontal. If μ_k = 0.15, determine the acceleration. Account for the vertical component of the applied force.
2. A 2 kg particle moves in a circle of radius 3 m at 6 m/s. Determine its normal acceleration and required inward resultant force. Explain which actual forces might provide this resultant.
3. A vehicle model has mass 1000 kg and experiences a constant 3000 N braking force plus quadratic drag with ρ = 1.2 kg/m³, C_D = 0.30, and A = 2.0 m². Formulate the differential equation for its speed and explain why constant-acceleration equations are not valid over the complete motion.

Multiple Choice Questions with Answers

Question 1

For a constant-mass particle, Newton’s second law is:

A. ∑F = mv

B. ∑F = ma

C. ∑F = m/a

D. ∑F = a/m

Correct Answer: B. ∑F = ma

Question 2

A 20 kg body has a resultant force of 60 N. Its acceleration is:

A. 0.33 m/s²

B. 3 m/s²

C. 40 m/s²

D. 1200 m/s²

Correct Answer: B. 3 m/s²

Question 3

In normal–tangential coordinates, normal acceleration is:

A. dv/dt

B. vρ²

C. v²/ρ

D. ρ/v²

Correct Answer: C. v²/ρ

Question 4

For uniform circular motion, the acceleration is:

A. Zero

B. Purely tangential

C. Purely inward toward the center

D. Always vertically downward

Correct Answer: C. Purely inward toward the center

Question 5

The limiting static-friction relation is:

A. F_s = 0 always

B. F_s ≤ μ_sN

C. F_s = μ_kN always

D. F_s = N/μ_s

Correct Answer: B. F_s ≤ μ_sN

Question 6

A 4 kg mass hangs at rest from a cable. Taking g = 9.81 m/s², the cable tension is:

A. 0.408 N

B. 4.00 N

C. 9.81 N

D. 39.24 N

Correct Answer: D. 39.24 N

Question 7

For an ideal linear spring, the magnitude of spring force is:

A. k/δ

B. kδ

C. δ/k

D. k + δ

Correct Answer: B. kδ

Question 8

Quadratic drag is commonly modelled as:

A. ½ρC_DAv²

B. ρC_DA/v²

C. cv with no velocity dependence beyond c

D. mg/v

Correct Answer: A. ½ρC_DAv²

Question 9

A 1200 kg vehicle travels at 10 m/s on a circular path of radius 50 m. Its normal resultant force is:

A. 240 N

B. 2400 N

C. 12,000 N

D. 60,000 N

Correct Answer: B. 2400 N

Question 10

Which statement about a free-body diagram is correct?

A. It includes only forces that produce motion.

B. It must include all external forces acting on the isolated body.

C. It combines forces acting on different bodies.

D. It excludes unknown reaction forces.

Correct Answer: B. It must include all external forces acting on the isolated body.