Advertisements
Advertisements
Question
A calorie is a unit of heat or energy and it equals about 4.2 J where 1J = 1 kg m2s–2. Suppose we employ a system of units in which the unit of mass equals α kg, the unit of length equals β m, the unit of time is γ s. Show that a calorie has a magnitude 4.2 α–1 β–2 γ2 in terms of the new units.
Advertisements
Solution 1
Given that,
1 calorie = 4.2 (1 kg) (1 m2) (1 s–2)
New unit of mass = α kg
Hence, in terms of the new unit, 1 kg =`1/alpha = a^(-1)`
In terms of the new unit of length,
`1m = 1/beta = beta^(-1) or 1m^2 = beta^(-2)`
And, in terms of the new unit of time,
`1s = 1y = y^(-1)`
`1s^2 = y^(-2)`
`1s^(-2) = y^2`
∴ 1 calorie = 4.2 (1 α–1) (1 β–2) (1 γ2) = 4.2 α–1 β–2 γ2
Solution 2
`n_2=n_1u_1/u_2=n_1([M_1^aL_1^bT_1^c])/([M_2^aL_2^bT_2^c])`
= n1 `[M_1/M_2]^a[L_1/L_2]^b[T_1/T_2]^c`
1 cal = 4.2 kg m2 s-2 ∴ a = 1, b = 2, c = -2
| SI | New System |
| `n_1 = 4.2` | `n_2 = ?` |
| `M_1 = 1 kg` | `M_2 = alpha kg` |
| `L_1 = 1m` | `L_2 = beta m` |
| `T_1 = 1 s` | `T_2 = y "second"` |
Now, n2 `=4.2[(1kg)/(alphakg)]^1[(1m)/(betam)]^2[(1s)/(gammas)]^(-2)`
`n_2 = 4.2 alpha^(-1) beta^(-2) gamma^2`
∴ 1 cal = `4.2 alpha^(-1) beta^(-2) gamma^2` in new system
APPEARS IN
RELATED QUESTIONS
The unit of length convenient on the atomic scale is known as an angstrom and is denoted by Å : 1Å = 10−10 m. The size of a hydrogen atom is about 0.5 Å. What is the total atomic volume in m3 of a mole of hydrogen atoms?
A physical quantity of the dimensions of length that can be formed out of c, G and `e^2/(4piε_0)` is (c is velocity of light, G is universal constant of gravitation and e is charge):
The dimensional formula for latent heat is ______.
If area (A), velocity (V) and density (p) are taken as fundamental units, what is the dimensional formula for force?
On the basis of dimensions, decide which of the following relations for the displacement of a particle undergoing simple harmonic motion is not correct ______.
- y = `a sin (2πt)/T`
- y = `a sin vt`
- y = `a/T sin (t/a)`
- y = `asqrt(2) (sin (2pit)/T - cos (2pit)/T)`
If P, Q, R are physical quantities, having different dimensions, which of the following combinations can never be a meaningful quantity?
- (P – Q)/R
- PQ – R
- PQ/R
- (PR – Q2)/R
- (R + Q)/P
Why length, mass and time are chosen as base quantities in mechanics?
Give an example of a physical quantity which has a unit but no dimensions.
Give an example of a physical quantity which has neither unit nor dimensions.
In the expression P = E l2 m–5 G–2, E, m, l and G denote energy, mass, angular momentum and gravitational constant, respectively. Show that P is a dimensionless quantity.
An artificial satellite is revolving around a planet of mass M and radius R, in a circular orbit of radius r. From Kepler’s Third law about the period of a satellite around a common central body, square of the period of revolution T is proportional to the cube of the radius of the orbit r. Show using dimensional analysis, that `T = k/R sqrt(r^3/g)`. where k is a dimensionless constant and g is acceleration due to gravity.
Einstein’s mass-energy relation emerging out of his famous theory of relativity relates mass (m ) to energy (E ) as E = mc2, where c is speed of light in vacuum. At the nuclear level, the magnitudes of energy are very small. The energy at nuclear level is usually measured in MeV, where 1 MeV= 1.6 × 10–13 J; the masses are measured in unified atomic mass unit (u) where 1u = 1.67 × 10–27 kg.
- Show that the energy equivalent of 1 u is 931.5 MeV.
- A student writes the relation as 1 u = 931.5 MeV. The teacher points out that the relation is dimensionally incorrect. Write the correct relation.
The entropy of any system is given by `S = alpha^2betaIn[(mukR)/(Jbeta^2) + 3]` Where α and β are the constants µ J, k, and R are no. of moles, the mechanical equivalent of heat, Boltzmann constant, and gas constant respectively. `["take S" = (dQ)/T]`
Choose the incorrect option from the following.
P = `alpha/beta` exp `(-"az"/"K"_"B"theta)`
θ `→` Temperature
P `→` Pressure
KB `→` Boltzmann constant
z `→` Distance
Dimension of β is ______.
A force defined by F = αt2 + βt acts on a particle at a given time t. The factor which is dimensionless, if α and β are constants, is ______.
