Physics
Physics
5th Edition
ISBN: 9781260486919
Author: GIAMBATTISTA
Publisher: MCG
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Chapter 1, Problem 91P

(a)

To determine

The combination of constants that has dimension of time.

(a)

Expert Solution
Check Mark

Answer to Problem 91P

The combination of constants that has dimension of time is hGc_.

Explanation of Solution

Write an expression for Plank’s time using the combinations of c, G, and h

tplanck=cpGqhr (I)

Here, c is the speed of light, G is the gravitational constant, h is the Planck’s constant, tplanck is the Planck’s time

p,q,r are integers. Rewrite the equation (I) in terms of units

s1=(ms)p(m3kg1s2)q(kg1m2s1)r (II)

Conclusion:

The power of base units should be same in both sides of equation (II). Consider the units separately.

For seconds,

1=p2qr (III)

For meters,

0=p+3q+2r (IV)

For kilogram,

0=0q+rq=r (V)

Substitute equation (V) in (III)

1=p2qq1=p3q (VI)

Substitute equation (V) in (IV)

0=p+3q+2q0=p+5qp=5q (VII)

Substitute 5q for p in equation (VI)

1=(5q)3qq=12

Substitute 12 for q in equation (V)

p=52

From equation (V)

q=r=12

Thus, according to the values of p,q,and r the only possible combination that has unit of time is

tplanck=hGc

Therefore, the combination of constants that has dimension of time is hGc_.

(b)

To determine

The Planck’s time in seconds.

(b)

Expert Solution
Check Mark

Answer to Problem 91P

The Planck’s time in seconds is tplanck=1.3×1043s_.

Explanation of Solution

Write the expression for Planck’s time

tplanck=hGc

Here, c is the speed of light, G is the gravitational constant, h is the Planck’s constant, tplanck is the Planck’s time

Conclusion:

Substitute 6.6×1034kgm2s1 for h, 6.7×1011m3kg1s2 for G , and 3.0×108m/s in above equation

tplanck=6.6×1034kgm2s1×6.7×1011m3kg1s23.0×108m/s=1.3×1043s

Therefore, the Planck’s time in seconds is tplanck=1.3×1043s_.

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