physics : class11 Dimensions of Constants : Dimensions of Physical Quantities : Physical World and Measurement MCQ Questions and Answers
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Dimensions of Constants
MCQ questions and answers.
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MCQ Questions
1. A physical displacement is represented by x = A sin(Bt) + C exp(-Dt), where t is time. If x has the dimensions of length, which set of dimensions is correct?
Explanation:
The arguments of sine and the exponential must be dimensionless, so [B][t] = 1 and [D][t] = 1, giving [B] = [D] = T^-1. Since both terms represent displacement, [A] = [C] = L.
2. In Newton's law of gravitation, F = Gm₁m₂/r². What are the dimensions of the gravitational constant G?
A.
M^-1 L^3 T^-2
B.
M L^3 T^-2
C.
M^-1 L^2 T^-2
D.
M^-2 L^3 T^-2
Correct Answer:
M^-1 L^3 T^-2
Explanation:
Rearranging gives G = Fr²/(m₁m₂). Using [F] = MLT^-2, [r²] = L², and [m₁m₂] = M², we obtain [G] = M^-1L^3T^-2.
3. The energy of a photon is given by E = hν. What are the dimensions of Planck's constant h?
A.
M L^2 T^-1
B.
M L^2 T^-2
C.
M L T^-1
D.
M^-1 L^2 T
Correct Answer:
M L^2 T^-1
Explanation:
Since h = E/ν, and energy has dimensions ML²T^-2 while frequency has dimensions T^-1, [h] = ML²T^-2/T^-1 = ML²T^-1.
4. A pressure-dependent relation is written as P = P₀ + a/V, where P is pressure and V is volume. What are the dimensions of the constant a?
A.
M L^-1 T^-2
B.
M L^2 T^-2
C.
M L^5 T^-2
D.
M^-1 L^2 T^2
Correct Answer:
M L^2 T^-2
Explanation:
The terms added to P must have the same dimensions, so [a/V] = [P]. Therefore [a] = [P][V] = (ML^-1T^-2)(L^3) = ML²T^-2.
5. In the expression v = at + b/t, where v is velocity and t is time, what are the dimensions of the product ab?
A.
L^2 T^-4
B.
L^2 T^-2
C.
L T^-2
D.
L^2 T^-3
Correct Answer:
L^2 T^-2
Explanation:
For at to have dimensions of velocity, [a] = LT^-2. For b/t to have dimensions of velocity, [b] = L. Hence [ab] = (LT^-2)(L) = L²T^-2.
6. The potential energy stored in a system is U = kx², where x is a displacement. What are the dimensions of the constant k?
A.
M L T^-2
B.
M T^-2
C.
M L^2 T^-2
D.
M L^-1 T^-2
Correct Answer:
M T^-2
Explanation:
Since k = U/x², and [U] = ML²T^-2 while [x²] = L², the dimensions of k are MT^-2.
7. A quantity is described by Q = Q₀ exp(-αt²), where Q and Q₀ have identical dimensions. What are the dimensions of the constant α?
A.
T^-2
B.
T^2
C.
1
D.
M T^-2
Correct Answer:
T^-2
Explanation:
The exponent must be dimensionless. Thus [α][t²] = 1, so [α] = T^-2.
8. The equation E = mc² + hν contains the constants c and h. What are the dimensions of the product h/c?
A.
M L T^-1
B.
M L T^-2
C.
M L^3 T^-1
D.
M T^-1
Correct Answer:
M L T^-1
Explanation:
The dimensions of h are ML²T^-1 and those of c are LT^-1. Therefore [h/c] = (ML²T^-1)/(LT^-1) = M L.
9. A constant k appears in the relation F = kx + p/t, where F is force, x is length, and t is time. If p has dimensions of momentum, what are the dimensions of k?
A.
M T^-2
B.
M L T^-2
C.
M L^-1 T^-2
D.
M T^-1
Correct Answer:
M T^-2
Explanation:
The first term kx must have dimensions of force. Hence [k] = [F]/[x] = (MLT^-2)/L = MT^-2. Also, p/t has dimensions (MLT^-1)/T = MLT^-2, confirming dimensional consistency.
10. The equation T = C l^a g^b relates a time period T to length l and acceleration due to gravity g. If a = 3/2 and b = -1/2, what are the dimensions of the constant C?
A.
L^-1
B.
1
C.
T L^-1
D.
L T^-1
Correct Answer:
L^-1
Explanation:
The dimensions of l^(3/2)g^(-1/2) are L^(3/2)(LT^-2)^(-1/2) = LT. Since [T] = T, the constant must have dimensions [C] = T/(LT) = L^-1.
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