We have calculated the centre of mass of a system of particles. How does it related to the movement of a system of particles? The centre of mass of a system of particles: $$\vec{r}_{CM} = \frac{\sum\limits_{i} m_{i}\vec{r}_{i}}{M}$$ The velocity of the centre of mass will be: $$\begin{aligned} \vec{v}_{CM} &= \frac{d \vec{r}_{CM}}{dt} \\ &= \frac{1}{M} \sum\limits_{i} m_{i} \frac{d\vec{r}_{i}}{dt} \\ &= \frac{\sum\limits_{i} …
Find the centre of mass of an uniform cone of height $h$ and radius $R$. Let the density of the cone be $\rho$. It is obvious from the diagram that the x and y components of the centre of mass of a cone is 0: $$\begin{aligned} x_{CM} &= 0 \\ y_{CM} &= 0 \end{aligned}$$ Hence, we just need to find …
Now, let’s get some practice on calculating centre of mass of objects. An object of mass $M$ is in the shape of a right-angle triangle whose dimensions are shown in the figure. Locate the coordinates of the centre of mass, assuming that the object has a uniform mass per unit area. Recall that the equations for centre of mass: $$\begin{aligned} …
We can associate a potential energy only to certain type of forces $F_{c}$. We call these forces conservative forces. $$\int\limits_{r_{1}}^{r_{2}} \vec{F}_{c}.d\vec{s} = W_{c} = U_{1}-U_{2} =-\Delta U$$ Gravitational force and the force that the spring exerts are both conservative forces. The energy that is “stored” as potential energy can be converted back to kinetic energy. Some forces are non-conservative. E.g. …
Kinetic Energy Kinetic energy is the energy associated with the motion of a body. $$K = \frac{1}{2} m v^{2}$$ Kinetic energy is a scalar and has the same units as work (J). The work done by a force F in displacing a particle is equal to the change in the kinetic energy of the particle. $$\begin{aligned} W &= K_{f}-K_{i} \\ …
In classical mechanics, we are concerned with the concepts of work and energy. Work-energy concepts are based on Newton’s laws and do not involve any new physical principles. The work-energy concept can therefore be applied to the dynamics of a mechanical system without resorting to Newton’s laws. This is useful in complex situation (e.g. variable forces) and applicable to a …
Uniform Circular Motion A particle moving with uniform speed $v$ in a circular path of radius $r$ experiences an acceleration $\vec{a}_{r}$ that has a magnitude: $$a_{r} = \frac{v^{2}}{r}$$ $\vec{a}_{r}$ is directed towards the center of the circle (centripetal acceleration) $\vec{a}_{r}$ is always perpendicular to $\vec{v}$ Applying Newton’s Second Law along the radial direction: $$\sum F_{r} = ma_{r} = m\frac{v^{2}}{r}$$ A …