4.5 Newton's Second Law

Newton's Second Law: Fundamental Relations

  • Acceleration (aa) is directly proportional to the applied force (FF) and inversely proportional to the mass (mm) of an object. This means that if the applied force on a given mass doubles, the acceleration will also double. Conversely, if the mass of an object doubles while the force remains constant, its acceleration will be halved.

  • Expressed as: a=F/ma = F/m or F=maF = ma (Equation 4.2 and 4.5).

  • The direction of acceleration ([a⃗][\vec{a}]) is always the same as the direction of the applied force ([F⃗][\vec{F}]). This means an object accelerates in the direction that it is pushed or pulled.

  • Expressed in vector form: [a⃗]=[F⃗]/m[\vec{a}] = [\vec{F}]/m or [F⃗]=m[a⃗][\vec{F}] = m[\vec{a}].

Net Force

  • When multiple forces ([F⃗]1,[F⃗]2,[F⃗]3,…[\vec{F}]_1, [\vec{F}]_2, [\vec{F}]_3, \dots) act on an object, the acceleration is determined by the net force ([F⃗]net[\vec{F}]_{\text{net}}).

  • The net force is the vector sum of all individual forces: [F⃗]net=[F⃗]1+[F⃗]2+[F⃗]3+…[\vec{F}]_{\text{net}} = [\vec{F}]_1 + [\vec{F}]_2 + [\vec{F}]_3 + \dots. This vector sum involves considering both the magnitude and direction of each force. For instance, if two forces act in opposite directions, the magnitude of the net force is the difference between their magnitudes, and its direction is that of the larger force.

  • Newton's Second Law with net force: [a⃗]=[F⃗]net/m[\vec{a}] = [\vec{F}]_{\text{net}}/m or [F⃗]net=m[a⃗][\vec{F}]_{\text{net}} = m[\vec{a}] (Equation 4.4).

  • The acceleration vector [a⃗][\vec{a}] points in the same direction as the net force vector [F⃗]net[\vec{F}]_{\text{net}} because mass (mm) is a scalar quantity and only scales the vector without changing its direction.

Unit of Force

  • The SI unit of force is the Newton (N).

  • One Newton is defined as: 1 N=1 kg⋅m/s21 \text{ N} = 1 \text{ kg} \cdot \text{m/s}^2.

  • Physically, one Newton is the amount of force required to accelerate a 1 kilogram (1 kg1 \text{ kg}) mass at a rate of 1 meter per second squared (1 m/s21 \text{ m/s}^2).

  • The English system unit for force is the pound (lb): 1 lb≈4.45 N1 \text{ lb} \approx 4.45 \text{ N}.

  • Newton's First and Second Laws are considered "laws of nature"