Physics · Chapter 4
Study notes aligned to the official NEB syllabus.
Dynamics is the branch of mechanics that studies the motion of a body while taking into account the cause of that motion, namely force. Unlike kinematics (which only describes motion), dynamics links motion to the forces that produce it, through Newton's laws.
This chapter builds the ideas of linear momentum and impulse, states Newton's laws in their momentum form, proves the principle of conservation of linear momentum, develops the turning effect of a force (moment and torque of a couple), states the conditions for equilibrium and the principle of moments, and finally treats the laws of solid friction and the angle of repose.
Newton's first law: Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled by some external force to change that state.
The first law gives a qualitative definition of force: force is the external agency that changes, or tends to change, the state of rest or of uniform motion of a body. It also introduces inertia.
Inertia is the inherent property of a body by virtue of which it cannot change its own state of rest or of uniform motion in a straight line by itself. There are three kinds:
Force is an external agency that changes or tends to change the state of rest or of uniform motion of a body. It is a vector quantity, denoted by $\vec{F}$. Its effects are to change the speed, the direction, the state (rest or motion), or the shape of a body. Its SI unit is the newton (N), where $1\ \text{N} = 1\ \text{kg m s}^{-2}$.
Newton's third law: To every action there is an equal and opposite reaction. Action and reaction are equal in magnitude and opposite in direction, but they act on two different bodies, which is why they do not cancel each other. Walking, swimming, and the recoil of a gun are everyday examples.
The second law is treated in Section 4, since it defines force through the rate of change of momentum.
Linear momentum of a body is the total quantity of motion contained in it, and is equal to the product of its mass and its velocity:
$$\vec{p} = m\vec{v}$$
It is a vector quantity directed along the velocity. Its SI unit is $\text{kg m s}^{-1}$ (equivalently $\text{N s}$), and its dimensional formula is $[\mathrm{M,L,T^{-1}}]$.
When a large force acts on a body for a very short time (as in a kick, a hammer blow, or a collision), the product of the force and the time for which it acts is called the impulse of the force.
$$\vec{J} = \vec{F},\Delta t$$
Impulse is a vector quantity in the direction of the force. Its SI unit is $\text{N s}$, and its dimensional formula is $[\mathrm{M,L,T^{-1}}]$, the same as that of momentum.
From Newton's second law, the force is the rate of change of momentum:
$$\vec{F} = \frac{d\vec{p}}{dt}$$
Rearranging and integrating over the time of action from $0$ to $t$:
$$\vec{F},dt = d\vec{p} \quad\Longrightarrow\quad \int_{0}^{t}\vec{F},dt = \int_{\vec{p}_1}^{\vec{p}_2} d\vec{p}$$
If the force is treated as constant (or as an average force $\vec{F}$) over the short interval $\Delta t$:
$$ \begin{aligned} \vec{F},\Delta t &= \vec{p}_2 - \vec{p}_1 = m\vec{v} - m\vec{u} \ \boxed{\ \vec{J} = \vec{F},\Delta t = \Delta \vec{p}\ } \end{aligned} $$
Impulse-momentum theorem: the impulse of a force is equal to the change in linear momentum it produces. This is why cricketers draw their hands back while catching a ball and why cars are fitted with airbags: increasing the time of impact $\Delta t$ reduces the force $\vec{F}$ for the same change in momentum.
Newton's second law: the force acting on a body is directly proportional to the rate of change of its momentum, and the change takes place in the direction of the applied force.
$$\vec{F} \propto \frac{d\vec{p}}{dt} \quad\Longrightarrow\quad \vec{F} = k,\frac{d\vec{p}}{dt}$$
where $k$ is a constant of proportionality. With SI units, $k = 1$, so:
$$\vec{F} = \frac{d\vec{p}}{dt}$$
Since $\vec{p} = m\vec{v}$ and the mass $m$ is constant:
$$\vec{F} = \frac{d(m\vec{v})}{dt} = m,\frac{d\vec{v}}{dt}$$
Using $\vec{a} = \dfrac{d\vec{v}}{dt}$:
$$\boxed{\ \vec{F} = m\vec{a}\ }$$
Thus the familiar equation $\vec{F} = m\vec{a}$ is a special case of the second law when the mass is constant. The second law is often called the real (or great) law of motion, because both the first law (put $\vec{F} = 0 \Rightarrow \vec{a} = 0$, so the body stays at rest or in uniform motion) and, together with interaction, the third law can be obtained from it.
Dynamics is the branch of mechanics that studies the motion of a body while taking into account the cause of that motion, namely force. Unlike kinematics (which only describes motion), dynamics links motion to the forces that produce it, through Newton's laws.
This chapter builds the ideas of linear momentum and impulse, states Newton's laws in their momentum form, proves the principle of conservation of linear momentum, develops the turning effect of a force (moment and torque of a couple), states the conditions for equilibrium and the principle of moments, and finally treats the laws of solid friction and the angle of repose.
Newton's first law: Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled by some external force to change that state.
The first law gives a qualitative definition of force: force is the external agency that changes, or tends to change, the state of rest or of uniform motion of a body. It also introduces inertia.
Inertia is the inherent property of a body by virtue of which it cannot change its own state of rest or of uniform motion in a straight line by itself. There are three kinds:
Force is an external agency that changes or tends to change the state of rest or of uniform motion of a body. It is a vector quantity, denoted by . Its effects are to change the speed, the direction, the state (rest or motion), or the shape of a body. Its SI unit is the newton (N), where .
Newton's third law: To every action there is an equal and opposite reaction. Action and reaction are equal in magnitude and opposite in direction, but they act on two different bodies, which is why they do not cancel each other. Walking, swimming, and the recoil of a gun are everyday examples.
The second law is treated in Section 4, since it defines force through the rate of change of momentum.
Linear momentum of a body is the total quantity of motion contained in it, and is equal to the product of its mass and its velocity:
It is a vector quantity directed along the velocity. Its SI unit is (equivalently ), and its dimensional formula is .
When a large force acts on a body for a very short time (as in a kick, a hammer blow, or a collision), the product of the force and the time for which it acts is called the impulse of the force.
Impulse is a vector quantity in the direction of the force. Its SI unit is , and its dimensional formula is , the same as that of momentum.
From Newton's second law, the force is the rate of change of momentum:
Rearranging and integrating over the time of action from to :
If the force is treated as constant (or as an average force ) over the short interval :
Impulse-momentum theorem: the impulse of a force is equal to the change in linear momentum it produces. This is why cricketers draw their hands back while catching a ball and why cars are fitted with airbags: increasing the time of impact reduces the force for the same change in momentum.
Newton's second law: the force acting on a body is directly proportional to the rate of change of its momentum, and the change takes place in the direction of the applied force.
where is a constant of proportionality. With SI units, , so:
Since and the mass is constant:
Using :
Thus the familiar equation is a special case of the second law when the mass is constant. The second law is often called the real (or great) law of motion, because both the first law (put , so the body stays at rest or in uniform motion) and, together with interaction, the third law can be obtained from it.