A pendulum consists of a tiny bob of mass M and a uniform cord of mass m and length L.
A) determine a formula for the period
B) what would be the fraction error if one used the formula for a simple pendulum.
A) To determine the formula for the period of the pendulum, we need to consider the factors that affect its motion. The period of a pendulum is the time it takes for one complete oscillation, which can be calculated using the formula:
T = 2π √(L/g)
where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity.
In this case, the pendulum consists of a bob (mass M) and a cord (mass m) of length L. The mass of the cord is assumed to be uniformly distributed along its length.
To account for the effect of the mass m of the cord, we need to find an equivalent effective length for the pendulum. The effective length can be calculated using the formula:
Leff = L + (m/3M) * L
where Leff is the effective length, M is the mass of the bob, and m is the mass of the cord.
Now, substituting Leff into the period formula, we have:
T = 2π √(Leff/g)
= 2π √((L + (m/3M) * L)/g)
Therefore, the formula for the period, considering the mass of the cord, is:
T = 2π √((L + (m/3M) * L)/g)
B) Now, let's calculate the fraction error if we were to use the formula for a simple pendulum (without considering the mass of the cord).
The formula for the period of a simple pendulum, neglecting the mass of the cord, is given by:
T_simple = 2π √(L/g)
To determine the fraction error, we need to compare T_simple (the value obtained using the simple pendulum formula) with the actual value of T (the value obtained using the formula accounting for the mass of the cord).
The fraction error can be calculated using the formula:
Fraction error = |(T - T_simple) / T|
Substituting the previously derived formulas into the fraction error formula, we have:
Fraction error = |(2π √((L + (m/3M) * L)/g) - 2π √(L/g)) / (2π √((L + (m/3M) * L)/g))|
By simplifying the expression, canceling out the common factors, and rearranging, we can calculate the fraction error.