Students can go through AP Inter 1st Year Physics Notes 7th Lesson Systems of Particles and Rotational Motion will help students in revising the entire concepts quickly.

## AP Inter 1st Year Physics Notes 7th Lesson Systems of Particles and Rotational Motion

→ The angle through which the radius vector rotates in a given time is called angular displacement.

→ The rate of angular displacement of a body is called angular velocity w = \(\frac{\mathrm{d} \theta}{\mathrm{dt}}\)

→ The rate of change of angular velocity of a body is called angular acceleration, a = \(\frac{\mathrm{d} \omega}{\mathrm{dt}}\)

→ A body which does not undergo any change in its shape and volume by the application of force is called a rigid body.

→ A pair of equal, unlike, parallel and non-collinear forces acting on a rigid body constitute a couple.

→ Uniform circular motion is the motion of a particle that moves on the circumference of a circle with constant angular velocity or constant linear speed.

→ Centripetal acceleration (a_{c}) is the acceleration of a particle moving on the circumference of a circle. Its magnitude is \(\frac{V^2}{r}\) (or) rω^{2} (or) vw. It is directed towards the centre of the circle and variable.

→ Centripetal force (F_{c}) is the force required by a particle to perform circular motion. Its magnitude is \(\frac{M v^2}{r}\) (or) Mrω^{2}. It is also directed towards the centre of the circle and variable. It is a real force like gravitational force, electrostatic force, force of friction etc. It is to be supplied to the particle by an external agency.

→ Safe maximum speed on an unbanked road is v = \(\sqrt{r g \mu}\).

→ A body performing horizontal circular motion has same speed at all points. When a stone tied to one end of a string is revolved in a horizontal circle,

The tension on the string = Centripetal force = \(\frac{M v^2}{r}\) = mrω^{2}

→ When a body performs rotational motion, each particle moves on a circle. The centers of all such circles are on a line called axis of rotation.

→ Torque or moment of force about a point is defined as the product of the force and the perpendicular distance of the point of application of the force from the point. In vector from, t = r × F.

→ Moment of inertia of a rigid body about an axis is defined as the sum of the products of the masses of different particles, supposed to be constituting the body and the square of their respective perpendicular distances from the axis of rotation.

→ Moment of inertia of a point mass m is I = mr^{2}, where r is the perpendicular distance of the point mass from the axis of rotation.

→ Radius of gyration of a rotating body is the distance between the axis of rotation and a point at which the whole mass of the body can be supposed to be concentrated so that its moment of inertia would be the same with the actual distribution of mass. Radius of gyration,

K = \(\sqrt{\frac{1}{M}}=\sqrt{\sum_{i=1}^n \frac{m_i r_i^2}{\text { Total mass }}}\)

→ Both moment of inertia and radius of gyration depend upon the position of axis of rotation and the distribution of mass about the axis of rotation. But moment of inertia depends on mass also.

→ Parallel axes theorem states that the moment of inertia of a rigid body about any axis is equal to the moment of inertia about a parallel axis through its center of mass plus the product of the mass the body and the square of the distance between the two parallel axes. Moment of inertia of a rigid body, I = I_{G} + Mr^{2}, where I_{G} is the moment of inertia of the body about a parallel axis through its center of mass and r is the distance between the parallel axis.

→ Perpendicular axes theorem states that the moment of inertia of plane lamina about an axis perpendicular to its plane is equal to the sum of the moments of inertia of the lamina about two axis perpendicular to each other, in its own plane and intersecting each other at the point, where the axis perpendicular to the plane passes.

→ Torque is the cause of angular acceleration. Relation between torque, t and angular acceleration, a is t = la.

→ Angular momentum of a particle about a point is the product of linear momentum of the particle and the perpendicular distance of the line of motion of the particle from the point. In vector form, L = r × p.

Its SI unit is kg m^{2} s^{-1} (or)Js. Its dimensional formula is [M L^{2}T^{-1}].

→ Angular momentum, L = mvr, where m is mass of a particle, v is the velocity and r is the perpendicular distance.

→ Angular momentum is expressed in terms of w as L = Iω.

→ Relation between τ and L is τ = \(\frac{\mathrm{dL}}{\mathrm{dt}}\) and τ and α is t = Iα.

→ Principle of conservation of angular momentum : When there is no resultant external torque on a rotating system, the angular momentum of the system remains constant both in magnitude and direction.

→ When there is no resultant external torque, L is constant i.e., Iω = constant (or) ω is inversely proportional to I.

→ When an object tied to one end of a string is revolved in a vertical circle, its velocity changes due to gravity.

→ The tension on the string at the highest point of a vertical circle = \(\frac{\mathrm{Mv}_2^2}{\dot{\mathrm{r}}}\) – Mg and at the lowest point, tension = \(\frac{\mathrm{Mv}_1^2}{\dot{\mathrm{r}}}\) + Mg. Tension at any position, T = \(\frac{M v^2}{r}\) – Mg cos θ.

→ The minimum velocity at the highest point, v_{2} = \(\sqrt{r g}\) and at the lowest point, v_{1} = \(\sqrt{5r g}\).

→ Centre of mass : Def: It is a point where the entire mass of a body or a system is supposed to be concentrated

→ Co-ordinates of Centre of mass :

X_{CM} = \(\frac{\Sigma m_i x_i}{\Sigma m_i}\); Y_{CM} = \(\frac{\Sigma m_i y_i}{\Sigma m_i}\); Z_{CM} = \(\frac{\Sigma m_i z_i}{\Sigma m_i}\)

→ Co-ordinates of velocity of C.M : V_{CM} = \(\frac{\Sigma m_i v_i}{\Sigma m_i}\)

→ Internal forces do not change the position (or) velocity of centre of mass.

→ Centre of mass obeys Newton’s laws of motion.

→ Matter may (or) may not be present at the centre of mass.

→ The linear momentum of,the centre of mass of a system is the sum of the linear momenta of all the particles comprising the system

Mv_{c} = Σ m_{i}v_{i} or P_{c} = P_{1} + P_{2} + …………….. P_{n}

M is the mass of the entire system, v_{c} in the velocity of the CM of the system. P_{c} in the momentum of centre of mass.

→ The acceleration imparted to the system to the external force F is equal to the acceleration of the centre of mass of the system.

a_{c} = \(\frac{\Sigma m_i z_i}{\Sigma m_i}\)

→ If a shell explodes in. mid air as it moves, the fragments of the shell move in different parabolic paths. but the centre of mass of the shell continues to move in the same parabolic path as the shell, as a single piece. would have moved.