Chapter 4: Linear Kinetics, Force and Newton’s Laws of Motion
Section Summary
4.1 Forces
- Dynamics is the study of how forces affect the motion of objects, whereas kinematics simply describes the way objects move.
- Force is a push or pull that can be defined in terms of various standards, and it is a vector that has both magnitude and direction.
- External forces are any outside forces that act on a body. A free-body diagram is a drawing of all external forces acting on a body.
- The SI unit of force is the newton (N).
4.2 Newton’s First Law
- According to Newton’s first law, there must be a cause for any change in velocity (a change in either magnitude or direction) to occur. This law is also known as the law of inertia.
- Friction is an external force that causes an object to slow down.
- Inertia is the tendency of an object to remain at rest or remain in motion. Inertia is related to an object’s mass.
- If an object’s velocity relative to a given frame is constant, then the frame is inertial. This means that for an inertial reference frame, Newton’s first law is valid.
- Equilibrium is achieved when the forces on a system are balanced.
- A net force of zero means that an object is either at rest or moving with constant velocity; that is, it is not accelerating.
4.3 Newton’s Second Law
- An external force acts on a system from outside the system, as opposed to internal forces, which act between components within the system.
- Newton’s second law of motion says that the net external force on an object with a certain mass is directly proportional to and in the same direction as the acceleration of the object.
- Newton’s second law can also describe net force as the instantaneous rate of change of momentum. Thus, a net external force causes nonzero acceleration.
4.4 Newton’s Third Law
- Newton’s third law of motion represents a basic symmetry in nature, with an experienced force equal in magnitude and opposite in direction to an exerted force.
- Two equal and opposite forces do not cancel because they act on different systems.
- Action-reaction pairs include a swimmer pushing off a wall, helicopters creating lift by pushing air down, and an octopus propelling itself forward by ejecting water from its body. Rockets, airplanes, and cars are pushed forward by a thrust reaction force.
- Choosing a system is an important analytical step in understanding the physics of a problem and solving it.
4.6 Mass and Weight
- Mass is the quantity of matter in a substance.
- The weight of an object is the net force on a falling object, or its gravitational force. The object experiences acceleration due to gravity.
- Some upward resistance force from the air acts on all falling objects on Earth, so they can never truly be in free fall.
- Careful distinctions must be made between free fall and weightlessness using the definition of weight as force due to gravity acting on an object of a certain mass.
4.7 Normal, Tension and Other Forces
- When an object rests on a surface, the surface applies a force to the object that supports the weight of the object. This supporting force acts perpendicular to and away from the surface. It is called a normal force.
- When an object rests on a nonaccelerating horizontal surface, the magnitude of the normal force is equal to the weight of the object.
- When an object rests on an inclined plane that makes an angle θθ with the horizontal surface, the weight of the object can be resolved into components that act perpendicular and parallel to the surface of the plane.
- The pulling force that acts along a stretched flexible connector, such as a rope or cable, is called tension. When a rope supports the weight of an object at rest, the tension in the rope is equal to the weight of the object. If the object is accelerating, tension is greater than weight, and if it is decelerating, tension is less than weight.
- The force of friction is a force experienced by a moving object (or an object that has a tendency to move) parallel to the interface opposing the motion (or its tendency).
- The force developed in a spring obeys Hooke’s law, according to which its magnitude is proportional to the displacement and has a sense in the opposite direction of the displacement.
- Real forces have a physical origin, whereas fictitious forces occur because the observer is in an accelerating or noninertial frame of reference.
4.8 Friction
- Friction is a contact force that opposes the motion or attempted motion between two systems. Simple friction is proportional to the normal force N supporting the two systems.
- The magnitude of static friction force between two materials stationary relative to each other is determined using the coefficient of static friction, which depends on both materials.
- The kinetic friction force between two materials moving relative to each other is determined using the coefficient of kinetic friction, which also depends on both materials and is always less than the coefficient of static friction.
4.9 Drag Forces
- Drag forces acting on an object moving in a fluid oppose the motion. For larger objects (such as a baseball) moving at a velocity [latex]v[/latex] in air, the drag force is given by
[latex]{F}_{\text{D}}=\frac{1}{2}{\mathrm{C\rho Av}}^{2},[/latex]
where [latex]C[/latex] is the drag coefficient (typical values are given in (Figure)), [latex]A[/latex] is the area of the object facing the fluid, and [latex]\rho[/latex] is the fluid density.
- For small objects (such as a bacterium) moving in a denser medium (such as water), the drag force is given by Stokes’ law,
[latex]{F}_{\text{s}}=6\text{πηrv},[/latex]
where [latex]r[/latex] is the radius of the object, [latex]\eta[/latex] is the fluid viscosity, and [latex]v[/latex] is the object’s velocity.
4.10 Elasticity: Stress and Strain
- Hooke’s law is given by
[latex]F=k\text{Δ}L,[/latex]
where [latex]\Delta L[/latex] is the amount of deformation (the change in length), [latex]F[/latex] is the applied force, and [latex]k[/latex] is a proportionality constant that depends on the shape and composition of the object and the direction of the force. The relationship between the deformation and the applied force can also be written as
[latex]\Delta L=\frac{1}{Y}\frac{F}{A}{L}_{0},[/latex]where [latex]Y\phantom{\rule{0.25em}{0ex}}[/latex] is Young’s modulus, which depends on the substance, [latex]A[/latex] is the cross-sectional area, and [latex]{L}_{0}[/latex] is the original length.
- The ratio of force to area, [latex]\frac{F}{A}[/latex], is defined as stress, measured in N/m2.
- The ratio of the change in length to length, [latex]\frac{\Delta L}{{L}_{0}}[/latex], is defined as strain (a unitless quantity). In other words,
[latex]\text{stress}=Y×\text{strain}.[/latex]
- The expression for shear deformation is
[latex]\Delta x=\frac{1}{S}\frac{F}{A}{L}_{0},[/latex]
where [latex]S[/latex] is the shear modulus and [latex]F[/latex] is the force applied perpendicular to [latex]{L}_{\text{0}}[/latex] and parallel to the cross-sectional area [latex]A[/latex].
- The relationship of the change in volume to other physical quantities is given by
[latex]\Delta V=\frac{1}{B}\frac{F}{A}{V}_{0},[/latex]
where [latex]B[/latex] is the bulk modulus, [latex]{V}_{\text{0}}[/latex] is the original volume, and [latex]\frac{F}{A}[/latex] is the force per unit area applied uniformly inward on all surfaces.
4.11 Problem-Solving Strategies
- To solve problems involving Newton’s laws of motion, follow the procedure described:
- Draw a sketch of the problem.
- Identify known and unknown quantities, and identify the system of interest. Draw a free-body diagram, which is a sketch showing all of the forces acting on an object. The object is represented by a dot, and the forces are represented by vectors extending in different directions from the dot. If vectors act in directions that are not horizontal or vertical, resolve the vectors into horizontal and vertical components and draw them on the free-body diagram.
- Write Newton’s second law in the horizontal and vertical directions and add the forces acting on the object. If the object does not accelerate in a particular direction (for example, the x-direction) then [latex]{F}_{\text{net}\phantom{\rule{0.25em}{0ex}}x}=0[/latex]. If the object does accelerate in that direction, [latex]{F}_{\text{net}\phantom{\rule{0.25em}{0ex}}x}=\text{ma}[/latex].
- Check your answer. Is the answer reasonable? Are the units correct?
4.12 Further Applications of Newton’s Laws
- Newton’s laws of motion can be applied in numerous situations to solve problems of motion.
- Some problems will contain multiple force vectors acting in different directions on an object. Be sure to draw diagrams, resolve all force vectors into horizontal and vertical components, and draw a free-body diagram. Always analyze the direction in which an object accelerates so that you can determine whether [latex]{F}_{\text{net}}=\text{ma}[/latex] or [latex]{F}_{\text{net}}=0[/latex].
- The normal force on an object is not always equal in magnitude to the weight of the object. If an object is accelerating, the normal force will be less than or greater than the weight of the object. Also, if the object is on an inclined plane, the normal force will always be less than the full weight of the object.
- Some problems will contain various physical quantities, such as forces, acceleration, velocity, or position. You can apply concepts from kinematics and dynamics in order to solve these problems of motion.