Units, Dimensions, and Error Analysis

Units, Dimensions and Errors

A vernier scale division is smaller than a main scale division in vernier callipers.
Vernier constant equals one main scale division multiplied by the number of vernier scale divisions.
The dimension of B²/µ₀, where B is magnetic field and µ₀ is permeability of free space, is [ML⁻¹T⁻²].
An expression defines the electric flux, ɸ = ασ + βλ, where σ and λ are linear and surface charge densities. α, therefore, represents displacement and β area.
If two vectors are perpendicular and represented by A = 2î + 3nĵ + 2k and B = 2î - 2ĵ + 4pk, then n = 3.
For E(t) = α³e⁻ᵝᵗ, where t is time and β = 0.3 s⁻¹, if uncertainties in α and t are 1.2% and 1.6% then find error in energy at t = 5s is 4%.
Matching:
- Permeability of free space MLT⁻²/A²
- Magnetic field ML⁰T⁻²A⁻¹
- Magnetic moment L² A
- Torsional constant ML²T⁻²
In y = (32.3 * 1125) / 27.4 with all digits significant, report y as 1330.
If the least count of a screw gauge is 0.01 mm, increasing the pitch by 75% and reducing circular scale divisions by 50% gives a new least count of 0.035 mm.
If E is energy, p is momentum, and c is light speed, the accurate equation is E² = p²c² + m²c⁴.
For a tiny rectangular sheet with length 5 mm and breadth 2.5 mm, the maximum fractional error in area using a screw gauge with a 0.75 mm pitch and 15 circular scale divisions is x / 100, where x is 3.
Measured quantity is the modulus of elasticity per unit torque, and the applied torque has dimension of [MᵃLTᵇ], If b = 3 then c = 0.
Pairs of physical quantities that do not share identical dimensions include pressure and Young's modulus.
In v = At² + Bt / (C + t), representing velocity (v) over time (t), the dimension of ABC is [ML²T⁻³].
Match the following:
- Young's modulus (II) ML⁻¹T⁻²
- Torque (III) ML²T⁻²
- Coefficient of viscosity (I) ML⁻¹T¹
- Gravitational constant (IV)M⁻¹L³T⁻²
Match the following:
- Magnetic induction (III) Gauss
- Magnetic intensity (IV) Ampere/meter
- Magnetic flux (I) Ampere meter²
- Magnetic moment (II) Weber
For Q = ab⁴ / cd where a=(60 ± 3)Pa, b=(20 ± 0.1)m; c=(40±0.2) Nsm⁻² and d=(50±0.1)m, the percentage error in Q is x/1000, where x = 7700.

Kinematics

A particle projected at 30° with 60 m/s has a height ratio of 5 between the first and last seconds before maximum height.
An airplane’s velocity-time graph allows the distance covered in 30.5 seconds to be determined as 9 km.
With a position vector of r = (5t²î - 5tĵ)m, the magnitude and direction of velocity at t=2s is 5√17 m/s at tan⁻¹ 4 in the -y direction.
An object's velocity-time graph indicates the distance covered from t=0 to t=4s is 10 m.
The ratio of maximum heights of two projectiles fired at the same initial speed at (45° – α) and (45° + α) is (1 - sin 2α) / (1 + sin 2α).
A boat travels downstream at 27 km/h in a river flowing at 9 km/h, and the range of a ball thrown upward at 10 m/s, observed from the river bank, is 2000 cm.
Two cars move in the same direction, where the first has a constant acceleration and the seconds initial accelaeration increases linearly time. Both cross the first time a time t=0 so maximum possible crossing in 3.

Laws of Motion

With a massless spring elongated by x₁ under tension 5N and x₂ under tension 7N, the tension for elongation (5x₁ – 2x₂) is 11 N.
For a car of mass 'm' on a banked road with radius 'r' and banking angle θ, the coefficient of friction to avoid slipping is µ = (v₀² - rg tan θ) / (rg + v₀² tan θ).
String length L is fixed at one end, carrying mass M, rotating 3/π times per second with tension (36)ML.
Release a balloon and content of mass M, which is moving up with constant acceleration 'a'. When ballon acceleration is '3a', the released mass is 2Ma/3a+g.

Work, Energy and Power

For a bob of mass m in vertical motion, suspended by a string of length l, the ratio of kinetic energy at points B and C is 2.
A force F=2î+bĵ+k is applied on a particle and it undergoes a displacement =î-2ĵ-k. To get the work done to be zero, the value of b is 1.
The decrease in kinetic energy of a 100g ball projected at 20 m/s at 60° from projection point to highest point is 15 J.
Body mass is 100g and velocity at point A is 10 m/s within circular radius 2m vertical direction. The ration between kinetic energies at points B and C is 4 : 3.
The work done against this force f=x²yî+y²ĵ on this particle of plane Ex+y=10 from point (0,0) to (4m,2m) is 152J.
A ball with kinetic energy KE is projected at an angle of 60° from the horizontal; its kinetic energy at the highest point of flight will be KE / 4.
If a force F = α + βx² acts on an object in the x-direction with work done 5J for a 1m displacement and a = 1N, then β will be 3N/m².
For a mass m sliding without friction down a vertical hoop of radius R, with a spring (constant k) at the bottom of the hoop (equilibrium length R) and v as velocity of bead, the combination is: 2√(gR + kR²/m).
Assertion: The value of network has an associated central force field is R because mechanics doesn’t have a potential energy
Under force F=(2î+3ĵ)N, a body of mass 4 kg is placed at a point P having coordinate (3, 4) m going to new point Q having coordinates (6, 10)m in 4 sec. The relation between average power and instantaneous power at the end of 4 sec is 1: 2.
Body of mass 'm' connected to a massless verticle circle of radius 'R' while gravity's other end of string under 9. Other end of the string is fixed constant acceleration due to gravity g. with velocity a top then find the ratio for kinetic energy for the circle is (n²+4)/n².

Center of Mass & Collision

A circular disc of radius 20 cm with its center at the origin with a circular hole cut out of the side. The distance from the center of mass of the center point of the origin 1cm.
Three spheres of mass undergo one dimensional motion, as shown to be both true, the light is not correct.
A sand dropper, drops sand of mass then determine how the force and the power relationship effect belt.

Rotation

A uniform circular disc of radius 'R' and mass 'M' with one quarter removed. The remaining moment of inertia is: 7MR²/32.
Given are the position vectors of two 1 kg particles, (A) and (B) respectively. At t = 1s (V) = (V) the value for L is: 90.
The torque due to the force is 3i - 3j.
Find the V² ratio is:1.

Gravitation

Small portion of mass (m) is the gravitational force to the center of the earth: 3/2 the center of mass of gravity is then calculated by gravitational constant (g).
Give a radius and mass that determines what the time period constant of the earth and moon are.
By calculating the angular time frame: .
Two planets determine with a mass of land is closed to: 2√2.

Properties of Solids

With volume 0.2%, the required pressure is approximately the volume of a cylinder is given by Nm^-2.
The volume of a solid copper: is 2 x [10^11].

Fluid Mechanics

Two soap bubbles with radius 2 and 4 cm in contact. The radius is 4.
A tube is filled, find the radius at cross-section in m squared. The velocity to a cylinder is used to gain the ability to find and calculate various points for a constant.

Simple Harmonic Motion

An assertion and pendulum, the light planets is given that the above and below true or false of statements are correct.
A light hollow cubed, solve for value y of value.
A particle period, solve by D and d of the correct value will it be there with this formula.