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Centre of mass formula

The formula for centre of mass are as stated below
Description Formula
Centre of mass of a system with n number of masses situated on a line at different positions   
Centre of mass of a system with n number of masses situated on a 2D plane
Centre of mass of a rectangular plate The centre of mass of a uniform rectangular plate of length L and breadth B is given as: r_x=B/2 r_y=L/2
Centre of mass of a triangular plate The centre of mass of a uniform triangular plate is given by the formula: r_c=h/3 Where, h is the height of the plate.
Centre of mass of a semi-circular ring The centre of mass of a semi-circular ring is given as: r_y=2R/π r_x=O Here, R is the radius of the semi- Circle.
Centre of mass of a semi-circular disc The centre of mass of a semi-circular disc is given as: r_y=4R/3π r_x=O Here, R is the radius of the semi- Circle.
Centre of mass of a hemispherical shell The centre of mass of a hemispherical shell is given as: r_y=R/2 r_x=O Here, R is the radius of the semi- Circle.
Centre of mass of a solid hemisphere The centre of mass of a solid hemisphere is given as: r_y=3R/8 r_x=O Here, R is the radius of the hemisphere.
Centre of mass of a circular cone The centre of mass of a circular cone is given as: r_y=h/4 Here, h is the height of the cone.
Centre of mass of a hollow circular cone The centre of mass of a hollow circular cone is given as: r_y=h/3 Here, h is the height of the cone.

Circular motion

The formula for circular motion are as stated below
Description Formula
Average angular velocity ω_average=(θ_2-θ_1)/(t_2-t_1 )
Average angular acceleration
Tangential acceleration a_t=dV/dt Here dV is the change in velocity over time dt. a_t=r dω/dt Here, r is the radius, dω is the change in angular frequency over time dt.
Centripetal acceleration a_c=v^2/r or a_c=ω^2 r Here, v is the linear velocity, r is the radius and ω is the angular frequency.
Normal reaction on a body moving on a concave bridge N=mg□cos cos θ +(mv^2)/r Here, m is the mass, g is the gravitational acceleration, θ is the angle, v is the linear velocity and r is the radius of the bridge.
Normal reaction on a convex bridge N=mg□cos cos θ -(mv^2)/r Here, m is the mass, g is the gravitational acceleration, θ  is the angle, v is the linear velocity and r is the radius.
Safe velocity of a vehicle on a level road v_safe≤√μgr Here, v safe is the safe velocity, is the coefficient of friction, g is the gravitational acceleration and r is the radius.
Banking angle tan θ =v^2/rg Here, θ  is the banking angle, v is the linear velocity, r is the radius of the curve and g is the gravitational acceleration.
Centrifugal force f=mω^2 r Here, f is the centrifugal force, m is the mass, is the angular velocity and r is the radius.
Conical pendulum T=2π√((L cos θ )/g) Here, L is the length of the pendulum, θ is the angle made by the string with the vertical and g is the gravitational acceleration.

De Broglie wavelength formula

The formula for de broglie wavelength are as stated below
Description Formula
De Broglie wavelength
Radius of electron in hydrogen like atoms r_n=n^2/Z a_0 Here, rn is the radius of nth orbit, a0 is a constant whose value is0.529×10^(-10) m m and z is the atomic number.
Speed of electron in hydrogen like atoms v_n=Z/n v_0 Here, Z is the atomic number, n is the orbit and v0 is a constant whose value is 2.19×106m/s.
Energy in nth orbit E_n=E_1⋅Z^2/n^2 Here,  En is energy of the nth orbit, E1 is the energy of the 1st orbit and its value is -13.6 eV, Z is the atomic number and n is the number orbit
Wavelength corresponding to spectral lines 1/λ=R[1/(n_1^2 )-1/(n_2^2 )]
Minimum wavelength for x rays λ_min=hc/(eV_0 ) Or λ_min=12400/V_0 ×10^(-10) m here, min is the minimum wavelength, h is the plank’s constant, c is the speed of light, e is the charge of an electron and V0 is the accelerating voltage.
Radius of nucleus R=R_0 A^(1/3) Here, R is the radius of the atom, R_0 is a constant whose value is 1.1×10^(-15)m, A is the mass number of the atom.
Number of nuclei during a radioactive decay N=N_0 e^(-λt)here, N is the number of nuclei at time t, N_0 is the initial number of nucleus and λ is the decay constant.
Half-life of a radioactive sample T_(1/2)=0.693/λ Here, T1/2 is the half-life period and  λ is the decay constant.
Average life T_av=T_(1/2)/0.693here, T_avis the average life and T1/2 is the half- life period.

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