A radioactive substance decays from 2.0kg to 1.6kg in 156 days.

What is the decay constant and the half life of the substance?

If T is the half life in days

(1.6)/2.0 = (T/156)^(1/2)
0.8 = 0.08006 T^1/2
9.992 = T^1/2
T = 99.8 days

The decay constant k is related to the half life by

k = ln2/T

A weight of 40 N is suspended from a spring that has a constant of 200 N/m the system is subjected to a harmonic driving force of frequency 10.5 hz resulting in a forced- motion amplitude of 3 cm determine the max value of the driving force

To find the decay constant and the half-life of a radioactive substance, we can use the exponential decay formula:

N(t) = N₀ * e^(-λt)

Where:
N(t) is the amount of the substance at time t
N₀ is the initial amount of the substance
λ is the decay constant
t is the time elapsed

Given that the substance decays from 2.0kg to 1.6kg in 156 days, we have:
N₀ = 2.0kg (initial amount)
N(t) = 1.6kg (amount after decay)
t = 156 days

First, we need to find the decay constant (λ).
Rearranging the formula, we can solve for λ:

λ = -ln(N(t) / N₀) / t

Substituting the given values, we get:
λ = -ln(1.6kg / 2.0kg) / 156 days

Calculating this expression, we find that λ ≈ 0.00722 (rounded to 5 decimal places).

Next, to find the half-life (t₁/₂), we use the equation:

t₁/₂ = ln(2) / λ

Substituting the value of λ, we get:
t₁/₂ = ln(2) / 0.00722

Evaluating this expression, we find that the half-life is approximately 95.89 days (rounded to 2 decimal places).