Q: The pressure P is related to distance x, Boltzmann constant k and temperature θ as P=bae−ax/kθ The dimensional formula of b is

The exponent is dimensionless. Hence[a]=[xkθ]=[mJK−1×K]=Jm−1=[ML2 T−2]×[L−1]=MLT−2Also [P]=[b][a]⇒[b]=[P][a]=[ML−1 T−2ML1 T−2]=[M0 L2 T0]

Q: The pressure P is related to distance x, Boltzmann constant k and temperature θ as P=bae−ax/kθ The dimensional formula of b is

The exponent is dimensionless. Hence[a]=[xkθ]=[mJK−1×K]=Jm−1=[ML2 T−2]×[L−1]=MLT−2Also [P]=[b][a]⇒[b]=[P][a]=[ML−1 T−2ML1 T−2]=[M0 L2 T0]

Q: The pressure P is related to distance x, Boltzmann constant k and temperature θ as P=bae−ax/kθ The dimensional formula of b is

The exponent is dimensionless. Hence[a]=[xkθ]=[mJK−1×K]=Jm−1=[ML2 T−2]×[L−1]=MLT−2Also [P]=[b][a]⇒[b]=[P][a]=[ML−1 T−2ML1 T−2]=[M0 L2 T0]

Q: The pressure P is related to distance x, Boltzmann constant k and temperature θ as P=bae−ax/kθ The dimensional formula of b is

The exponent is dimensionless. Hence[a]=[xkθ]=[mJK−1×K]=Jm−1=[ML2 T−2]×[L−1]=MLT−2Also [P]=[b][a]⇒[b]=[P][a]=[ML−1 T−2ML1 T−2]=[M0 L2 T0]

Q: The magnitude of induced emf e in a conductor of length L rotating in magnetic field B is given by e=a1(πBL2) The dimensional formula of a is

a=eπBL2=[ML2 T−3 A−1][M0 L0 T0]×[ML0 T−2 A−1]×[L2]=[M0 L0 T1]

Q: Two resistors R1=[3.0±0.1]Ω and R2=(6.0± 0.3) Ω are connected in parallel. The resistance of the combination is

R1=R11+R21⇒R=R1+R2R1R2=9.03.0×6.0=2.0Ω∴+R2δR=+R12δR1+R22δR2=(3)20.1+(6)20.3=0.019∴δR=0.019×R2=0.019×(2)2=0.076≃0.08Ω

Q: Two resistors R1=[3.0±0.1]Ω and R2=(6.0± 0.3) Ω are connected in series, the maximum percentage error in the resistance of the combination will be

R=R1+R2=3.0+6.0=9.0ΩδR=δR1+δR2=0.1+0.3=0.4∴ Maximum percentage error =RδR×100=9.00.4×100=4.4%

Q: Which of the following pairs of physical quantities do not have the same dimensions?

Pressure: Area force Stress: Area force Youngs modulus: strain stress , here strain is dimension less So, all three has same dimension.Electromotive force: charge Energy Potential difference: QW, with is same as ElectromotiveElectric voltage: If you see in a capacitor, then Voltage is given by charge Energy Heat: ML2 T−2Potential energy: mgh⇒M2 T−2Work done =f.d⇒ML2 T−2Electric flux: M−2SAElectric field: M2 S−3 A−1

Q: Which of the following pairs of physical quantities have different dimensions?

Using L=mvr=2πnhThus the angular momentum (L) has the same dimensions as that of Planck's constant (h).Impulse I = ΔPThus impulse (I) has the same dimensions as that of linear momentum (P)Using τ=∣a⇒∣=aτwhere angular acceleration is not dimensionless.Thus the dimensions of moment of inertia are different from those of moment of force (OR torque)Dimension of energy E=[ML2 T−2]Dimension of torque τ=Fr=ML2 T−2Hence, torque and energy have same dimensions.

Question 1

The pressure P is related to distance x, Boltzmann constant k and temperature θ as P=ba​e−ax/kθ The dimensional formula of b is

Accepted Answer

The exponent is dimensionless. Hence[a]=[xkθ​]​=[mJK−1×K​]=Jm−1=[ML2 T−2]×[L−1]=MLT−2​Also [P]=[b][a]​⇒[b]=[P][a]​=[ML−1 T−2ML1 T−2​]=[M0 L2 T0]

Question 2

The pressure P is related to distance x, Boltzmann constant k and temperature θ as P=ba​e−ax/kθ The dimensional formula of b is

Accepted Answer

The exponent is dimensionless. Hence[a]=[xkθ​]​=[mJK−1×K​]=Jm−1=[ML2 T−2]×[L−1]=MLT−2​Also [P]=[b][a]​⇒[b]=[P][a]​=[ML−1 T−2ML1 T−2​]=[M0 L2 T0]

Question 3

The pressure P is related to distance x, Boltzmann constant k and temperature θ as P=ba​e−ax/kθ The dimensional formula of b is

Accepted Answer

The exponent is dimensionless. Hence[a]=[xkθ​]​=[mJK−1×K​]=Jm−1=[ML2 T−2]×[L−1]=MLT−2​Also [P]=[b][a]​⇒[b]=[P][a]​=[ML−1 T−2ML1 T−2​]=[M0 L2 T0]

Question 4

The pressure P is related to distance x, Boltzmann constant k and temperature θ as P=ba​e−ax/kθ The dimensional formula of b is

Accepted Answer

The exponent is dimensionless. Hence[a]=[xkθ​]​=[mJK−1×K​]=Jm−1=[ML2 T−2]×[L−1]=MLT−2​Also [P]=[b][a]​⇒[b]=[P][a]​=[ML−1 T−2ML1 T−2​]=[M0 L2 T0]

Question 5

The magnitude of induced emf e in a conductor of length L rotating in magnetic field B is given by e=a1​(πBL2) The dimensional formula of a is

Accepted Answer

a=eπBL2​​=[ML2 T−3 A−1][M0 L0 T0]×[ML0 T−2 A−1]×[L2]​=[M0 L0 T1]​

Question 6

Two resistors R1​=[3.0±0.1]Ω and R2​=(6.0± 0.3) Ω are connected in parallel. The resistance of the combination is

Accepted Answer

R1​=R1​1​+R2​1​⇒R=R1​+R2​R1​R2​​=9.03.0×6.0​=2.0Ω∴+R2δR​=+R12​δR1​​+R22​δR2​​=(3)20.1​+(6)20.3​=0.019∴δR=0.019×R2=0.019×(2)2=0.076≃0.08Ω

Question 7

Two resistors R1​=[3.0±0.1]Ω and R2​=(6.0± 0.3) Ω are connected in series, the maximum percentage error in the resistance of the combination will be

Accepted Answer

R=R1​+R2​=3.0+6.0=9.0ΩδR=δR1​+δR2​=0.1+0.3=0.4∴ Maximum percentage error ​=RδR​×100=9.00.4​×100=4.4%​

Question 8

Which of the following pairs of physical quantities do not have the same dimensions?

Accepted Answer

Pressure:  Area  force ​Stress:  Area  force ​Youngs modulus:  strain  stress ​, here strain is dimension less So, all three has same dimension.Electromotive force:  charge  Energy ​Potential difference: QW​, with is same as ElectromotiveElectric voltage: If you see in a capacitor, then Voltage is given by  charge  Energy ​Heat: ML2 T−2Potential energy: mgh⇒M2 T−2Work done =f.d⇒ML2 T−2Electric flux: M−2SAElectric field: M2 S−3 A−1

Question 9

Which of the following pairs of physical quantities have different dimensions?

Accepted Answer

Using L=mvr=2πnh​Thus the angular momentum (L) has the same dimensions as that of Planck's constant (h).Impulse I = ΔPThus impulse (I) has the same dimensions as that of linear momentum (P)Using τ=∣a⇒∣=aτ​where angular acceleration is not dimensionless.Thus the dimensions of moment of inertia are different from those of moment of force (OR torque)Dimension of energy E=[ML2 T−2]Dimension of torque τ=Fr=ML2 T−2Hence, torque and energy have same dimensions.

Question 10

In the expression A=A0​e−a/kT,k is Boltzmann constant and T is the absolute temperature. The dimensions of a are the same as those of

Accepted Answer

Since kT has dimensions of energy, the correct choice is (a).

Question 11

A cube has a side of 1.2 cm. The volume of the cube up to appropriate significant figures is

Accepted Answer

V=L3​=1.2 cm×1.2 cm×1.2 cm=1.728 cm3​Since there are two significant figures in L=1.2 cm, the volume is accurate only up to two significant figure.

Question 12

The quantities L/R and RC (where L,C and R stand for inductance, capacitance and resistance respectively) have the same dimensions as those of

Accepted Answer

L/R is the time constant of an L−R circuit and CR is the time constant of a C−R circuit. The dimension of the time constant is the same as that of time.

Question 13

Which one of the following has the dimensions of ML−1 T−2 ?

Accepted Answer

ML−1 T−2 are the dimensions of force per unit area. Out of the four choices, stress is the only quantity that is force per unit area.

Question 14

The dimensions of angular momentum are

Accepted Answer

The angular momentum L of a particle with respect to point whose position vector is r is given byL=r×pwhere p is the linear momentum of the moving particle.∴ Dimensions of L= dimension of r× dimensions of p=L×MLT−1=ML2 T−1

Question 15

According to the quantum theory, the energy E of a photon of frequency v is given by E=hv where h is Planck's constant. What is the dimensional formula for h ?

Accepted Answer

Dimensions of h= dimension of v dimension of E​= T−1ML2 T−2​ =ML2 T−1

Question 16

The dimensions of Planck's constant are the same as those of

Accepted Answer

Using L=2πnh​where L is angular momentum and h is Planck's constant As n/2π is just a number, thus dimensions of Planck's constant is same as that of angular momentum.

Question 17

The volume V of water passing any point of a uniform tube during t seconds is related to the cross-sectional area A of the tube and velocity u of water by the relation V∝Aαuβtγ which one of the following will be true?

Accepted Answer

The relation is given byV∝Aαuβtγ[M0 L3 T0]=[L2αLβT−βTγ]By equating the coefficient,γ−β=0γ=β……(1)and 2α+β=3α=β=γ

Question 18

The volume V of water passing any point of a uniform tube during t seconds is related to the cross-sectional area A of the tube and velocity u of water by the relation V∝Aαuβtγ which one of the following will be true?

Accepted Answer

The dimensions of the two sides of proportionality areL3=L2α(LT−1)βTγ=L2α+βTγ−βEquating the powers of dimensions on both sides, we have2α+β=3γ−β=0which give β=γ and α=21​(3−β), i.e. α=β=γ

Question 19

The frequency n of vibrations of uniform string of length l and stretched with a force F is given by n=2lp​mF​​ where p is the number of segments of the vibrating string and m is a constant of the string. What are the dimensions of m ?

Accepted Answer

Squaring both sides of the given relation, we getn2=4l2p2​⋅mF​ or m=4l2n2p2F​∴ dimensions of m= dimensions of l2× dimensions of n2 dimensions of F​( ∵p is a dimensionless number)= L2×(T−1)2MLT−2​=ML−1 T0

Question 20

If velocity (V), acceleration (A) and force (F) are taken as fundamental quantities instead of mass (M), length (L) and time (T), the dimensions of Young's modulus would be

Accepted Answer

Dimensions of Young's modulus Y are ML−1 T−2. The dimension of V,A and F in terms of M,L and T are(V)=LT−1,(A)=LT−2 and (F)=MLT−2 Let (Y)=(VaAbFc)​Putting dimensions of Y,V,A and F. We have(ML−1 T−2)=(LT−1)a×(LT−2)b×(MLT−2)corM1 L−1 T−2=Mc La+b+c T−a−2b−2cEquating powers of M,L and T we havec=1,a+b+c=−1and −a−2b−2c=−2which give a=−4,b=2 and c=1.Thus (Y)=(FA2V−4)

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Solutions for IIT JEE Advanced Comprehensive Physics by N.K.Bajaj

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