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Worksheet answers solution
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(a) A = πr^2 d = 2r r = d 2
A = π
[ (^) d 2
] 2
A = π d
2 4 (X ± ∆X)^2 = (X)^2 ± 2 X∆X Using the above equation on page 7 of the lab manual How? (X ± ∆X)^2 = (X)^2 ± 2 X∆X We can expand (X ± ∆X)^2 = X^2 ± 2 X∆X + ∆X^2 ∆X^2 is too small, therefore we can ignore the term ∆X^2 (X ± ∆X)^2 = X^2 ± 2 X∆X (This is same as the equation in page 7 of the lab manual) ∆(d^2 ) = 2d∆d ∆A = π 4 2 d∆d ∆A = π 2 d∆d A = πd
2 4 =^
π(1.22)^2 4 = 1.^168 mm
2
1
∆A = π 2 d∆d = π 2 × 1. 22 × 0 .02 = 0. 038 mm^2 A = 1. 168 ± 0 .038mm^2 OR alternatively using the following method which may be easier than previous method! ∆A A =
2∆d d How? ∆A = π 4 2 d∆d A = π d
2 4 Therefore (^) ∆A A =
π 4 2 d∆d π d 42 ∆A A =
2∆d d ∆A = A × 2 × ∆ dd =^1.^168 × 1.^222 × 0.^02 = 0.038mm^2 (b) T =^1 f Using the following equation on page 7 of the lab manual (X ± ∆X)n^ = (X)n^ ± nXn−^1 ∆X f = T^1 = T −^1 where n=- ∆f = − 1 × T −^1 −^1 × ∆T ∆f = − 1 × T −^2 × ∆T = − T∆ 2 T Magnitude of the error is given by ∆f = ∆ TT 2 OR Using the following equation ∆f f =
∆f = f × ∆ TT = T^1 × ∆ TT = ∆ TT 2 T = 2. 3 ± 0. 1 s f = T^1 = (^21). 3 = 0. 435 Hz ∆f = ∆ TT 2 = 20 .. 312 = 0. 019 s f = 0. 44 ± 0. 02 Hz 2
(f) λ = d sinθ ∆λ λ =
√√ √√^ (^ ∆d d
) 2
( (^) ∆(sinθ) sinθ
) 2
∆λ λ =
√√√ √
( (^) ∆d d
) 2
( (^) cosθ∆θ sinθ
) 2
∆λ λ =
√√√ √
( (^) ∆d d
) 2
( (^) ∆θ tanθ
) 2
∆λ = λ
√√ √√^ (^ ∆d d
) 2
( (^) ∆θ tanθ
) 2
∆λ = d sinθ
√√ √√^ (^ ∆d d
) 2
( (^) ∆θ tanθ
) 2
d = 1. 67 ± 0. 03 μm θ = 17. 42 ± 0. 07 o λ = d sinθ = 1. 67 × sin 17 .42 = 0. 499954 μm ∆θ = 0. 07 o^ = 570 ..^07293 rad
∆λ = 0. 499954 ×
√(
) 2
) 2 = 9. 189 nm λ = 499. 954 nm λ = 500 ± 9 nm
g′ = g
( 1 − (^) R^2 hE
)
g′ = g − (^2) RhgE Comparing the above equation with y = mx + c c = g m = − (^) R^2 gE
Value of the gradient (m ± ∆m) and intercept (c ± ∆c) are at the bottom of the graph (next page). This graph is plotted by using LineFit. In addition, LineFit also calculate the gradient and intercept (with errors) for the best fit. This can not be done by using Excel. However Excel will not calculate the errors in gradient or intercept. These values can be calculated using the equations on page 20 of the lab manual!
m = − 2. 87 × 10 −^3 ± 1. 10 × 10 −^4 c = 9. 79 ± 6. 83 × 10 −^3 Gradient = R^2 gE RE = (^) Gradient^2 g =^2 × Gradient^ Intercept RE = (^2).^287 × ×^9. 1079 − 3 = 6. 822 × 103 km ∆RE RE^ =
√( ∆C C
) 2
( (^) ∆m m
) 2 = 0. 03833 ∆RE = RE × 0 .03833 = 261km RE = 6822 ± 261 km
4 9 14 19 24 29 34 39 44 49
-1.
-1.
-0.
-0.
Time(sec)
ln (V ol ta ge )
Fit Results Goodness of fit = 7.57e- Slope M = -1.00e-01 +/- 1.53e- Intercept C = 2.89e+00 +/- 1.42e-
