Physics in Depth

A particle is in uniform circular motion if it travels around a circle on a circular arc at uniform (constant) speed. Although the speed does not vary , the particle is accelerating because the velocity changes in direction. fig. below shows the relationship between the velocity and acceleration vectors at various stages during uniform circular motion. Both vectors have constant magnitude, but their directions change continuously. The velocity is always directed tangent to the circle in the direction of motion. The acceleration is always directed radially inward. Because of this, the acceleration associated with uni form circular motion is called a centripetal (meaning "center seeking") acceleration. Centripetal acceleration Consider a particle 'p' moves at constant speed 'v' around a circle of radius r. At the instant shown in figure above, p has a coordinate x and y. \(\theta\) be the angular displacement of p. ...

Work, Energy and Power Work Work is said to be done by a force when the force produces a displacement on a body on which it acts. \[W=\vec{F}.\vec{S}\] Here, W is the work done by the force \(\vec{F}\) while displacing a body through \(\vec{s}\). Work is a scalar quantity; it has no property of direction but only magnitude. If the angle between the displacement vector and force vector is \(\theta\) as in fig., then the work done is, \[W=Fscos\theta \hspace{0.1cm}(\because \vec{a}.\vec{b}=abcos\theta)\] Equivalently, W= component of force along the displacement \(\times\) the displacement Special cases : When \(\theta=0^\circ\), W=Fs ( maximum work done ). When \(\theta=90^\circ\), W=0 ( no work done ) When \(\theta positive work done ) When \(90^\circ negative work done ) \(\implies\) When a coolie travels on a platform with a load on his head, he exerts a vertical force on the load as in fig. . In this case, the angle \(\theta\) between th...

Mass and Pulley system Equation of motion for mass m is, \[\begin{align} F&=T-mg \nonumber \\ ma&=T-mg ... (i) \end{align}\] Equation of motion for mass M is, \[\begin{align} F&=T-Mg \nonumber \\ Ma&=T-Mg ... (ii) \end{align}\] Adding eqns (i) and (ii), \[\begin{align*} ma+Ma&=Mg-mg\\ a&=\frac{(M-m)g}{(M+m)} ... (iii) \end{align*}\] Substituting the vaue of a in eqn (i), \[\begin{align*} m\left(\frac{(M-m)g}{M+m}\right)&=T-mg\\ \frac{m(M-m)g}{M+m}+mg&=T\\ T&=\frac{m(M-m)g+mg(M+m)}{M-m}\\ \therefore T&=\frac{2Mm}{M+m}g ... (iv) \end{align*}\] Two masses 7 kg and 12 kg are connected at the two ends of a light inextensible string that passes over a frictionless pulley. Using free body diagram method, find the acceleration of masses and the tension in the string, when the masses are released. Here, m = 7 kg and M = 12 kg. Prior to using the formula directly...

Newton's Laws of Motion Image source : Britannica In 1687, Newton formulated his laws of motion and published it in his book Philosophiae Naturalis Principia Mathematica . Isaac Newton first understood the relation between a force and the acceleration and study of this relation is given a name , Newtonian Mechanics . Though, there are some restrictions to Newtonian Mechanics such as: If the speeds of the interacting bodies are very large - an appreciable fraction of the speed of light- we must replace Newtonian mechanics with Einstein's special theory of relativity. If the interacting bodies are on the scale of atomic structure (for example, electrons) we must replace Newtonian mechanics with quantum mechanics. , it still applies to the motion of objects ranging in size from the very small (almost on the scale of atomic structure) to astronomical (galaxies and clusters of galaxies). Since, theory of relativity and quantum mechanics are beyond the...

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Relative velocity Relative velocity, \(\vec{v_{AB}}\), is the velocity of an object A in the rest frame of an object B. Rest frame of a particle is the coordinate system (frame of reference) in which the particle is at rest . It can be simply defined as, the velocity of an object with respect to another object . Suppose you are in a car A moving north at 90 km/h (with respect to the object in the ground outside). To another man in a car B moving towards north with same speed, you seem to be stationary. Let's discuss this in more detail. For this, we consider following three cases: When the two bodies, body A and body B, are moving in the same direction with velocity \(\vec{v_A}\) and \(\vec{v_B}\) respectively, then the relative velocity of A with respect to B is, \[\vec{v_{AB}}=\vec{v_A}-\vec{v_B}\] So, in above example, it must be clear that for a person in car B moving alongside you with the same speed as yours, you are at rest. When this bodies ar...

Kinematics Rest and Motion When do you declare that some body is in rest or in motion ? Place the book on a table and watch it! Is it in rest? Imagine you being on moon and by some means you could watch your book on your table. Would you again see it in rest? In first case, you would see that the book is in rest but when you are on moon you would see the table, the book and the Earth as a whole in a motion. You are correct in both the cases and it's a good news. Hurray! .It is because rest and motion are not absolute terms but are relative . The book is in rest when you see it from your room because you and the book both do not change position with respect to each other, but it is in motion when you see it from the moon because the moon is moving with respect to the book and book moves with respect to the moon. So, let's set up a definition. If a body changes it's position with respect to the observer ( or reference frame ), then it is said t...