# Mechanics: Newton's Laws of Motion

## Newton's Laws of Motion: Problem Set Overview

This set of 30 problems targets your ability to distinguish between mass and weight, determine the net force from the values of the individual forces, relate the acceleration to the net force and the mass, analyze physical situations to draw a free body diagram and solve for an unknown quantity (acceleration or individual force value), and to combine a Newton's second law analysis with kinematics to solve for an unknown quantity (kinematic quantity or a force value). Problems range in difficulty from the very easy and straight-forward to the very difficult and complex. The more difficult problems are color-coded as blue problems.

## Mass versus Weight

This set of 30 problems targets your ability to distinguish between mass and weight, determine the net force from the values of the individual forces, relate the acceleration to the net force and the mass, analyze physical situations to draw a free body diagram and solve for an unknown quantity (acceleration or individual force value), aMass is a quantity which is dependent upon the amount of matter present in an object; it is commonly expressed in units of kilograms. Being the amount of matter possessed by an object, the mass is independent of its location in the universe. Weight, on the other hand, is the force of gravity with which the Earth attracts an object towards itself. Since gravitational forces vary with location, the weight of an object on the Earth's surface is different than its weight on the moon. Being a force, weight is most commonly expressed in the metric unit as Newtons. Every location in the universe is characterized by a gravitational field constant represented by the symbol g (sometimes referred to as the acceleration of gravity). Weight (or Fgrav) and mass (m) are related by the equation:

Fgrav = m • g

## Newton's Second Law of Motion

Newton's second law of motion states that the acceleration (a) experienced by an object is directly proportional to the net force (Fnet) experienced by the object and inversely proportional to the mass of the object. In equation form, it could be said that a = Fnet/m. The net force is the vector sum of all the individual force values. If the magnitude and direction of the individual forces are known, then these forces can be added as vectors to determine the net force. Attention must be given to the vector nature of force. Direction is important. An up force and a down force can be added by assigning the down force a negative value and the up force a positive value. In a similar manner, a rightward force and a leftward force can be added by assigning the leftward force a negative value and the rightward force a positive value.

The a = Fnet/m equation can be used as both a formula for problem solving and as a guide to thinking. When using the equation as a formula for problem solving, it is important that numerical values for two of the three variables in the equation be known in order to solve for the unknown quantity. When using the equation as a guide to thinking, thought must be given to the direct and inverse relationships between acceleration and the net force and mass. A two-fold or a three-fold increase in the net force will cause the same change in the acceleration, doubling or tripling its value. A two-fold or three-fold increase in the mass will cause an inverse change in the acceleration, reducing its value by a factor of two or a factor of three.

## Free Body Diagrams

Free body diagrams represent the forces which act upon an object at a given moment in time. The individual forces which act upon an object are represented by vector arrows. The direction of the arrows indicate the direction of the force and the approximate length of the arrow represents the relative magnitude of the force. The forces are labeled according to their type. A free body diagram can be a useful aid in the problem-solving process. It provides a visual representation of the forces exerted upon an object. If the magnitudes of all the individual forces are known, the diagram can be used to determine the net force. And if the acceleration and the mass are known, then the net force can be calculated and the diagram can be used to determine the value of a single unknown force.

## Coefficient of Friction

An object which is moving (or event attempting to move) across a surface encounters a force of friction. Friction force results from the two surfaces being pressed together closely, causing intermolecular attractive forces between molecules of different surfaces. As such, friction depends upon the nature of the two surfaces and upon the degree to which they are pressed together. The friction force can be calculated using the equation:

Ffrict = µ• Fnorm

The symbol µ (pronounced "mew") represents of the coefficient of friction and will be different for different surfaces.

## Blending Newton's Laws and Kinematic Equations

Kinematics pertains to a description of the motion of an object and focuses on questions of how far?, how fast?, how much time? and with what acceleration? To assist in answering such questions, four kinematic equations were presented in the One-Dimensional Kinematics unit. The four equations are listed below.

• d = vo • t + 0.5 • a • t2
• vf = vo + a • t
• vf2 = vo 2 + 2 • a • d
• d = (vo + vf)/ 2 • t

where

• d = displacement
• t = time
• a = acceleration
• vo = original or initial velocity
• vf = final velocity

Newton's laws and kinematics share one of these questions in common: with what acceleration? The acceleration (a) of the Fnet = m•a equation is the same acceleration of the kinematic equations. Common tasks thus involve:

1. using kinematics information to determine an acceleration and then using the acceleration in a Newton's laws analysis, or
2. using force and mass information to determine an acceleration value and then using the acceleration in a kinematic analysis.

When analyzing a physics word problem, it is wise to identify the known quantities and to organize them as either kinematic quantities or as F-m-a type quantities.

## Habits of an Effective Problem-Solver

An effective problem solver by habit approaches a physics problem in a manner that reflects a collection of disciplined habits. While not every effective problem solver employs the same approach, they all have habits which they share in common. These habits are described briefly here. An effective problem-solver...

• ...reads the problem carefully and develops a mental picture of the physical situation. If needed, they sketch a simple diagram of the physical situation to help visualize it.
• ...identifies the known and unknown quantities in an organized manner, often times recording them on the diagram iteself. They equate given values to the symbols used to represent the corresponding quantity (e.g., vo = 0 m/s, a = 2.67 m/s/s, vf = ???).
• ...plots a strategy for solving for the unknown quantity; the strategy will typically center around the use of physics equations be heavily dependent upon an understaning of physics principles.
• ...identifies the appropriate formula(s) to use, often times writing them down. Where needed, they perform the needed conversion of quantities into the proper unit.
• ...performs substitutions and algebraic manipulations in order to solve for the unknown quantity.