A child plays on a bungee cord and oscillates with a certain frequency f. An adult with a mass that is 16 times greater than that of the child then uses the same bungee cord. What is the ratio of the frequency with the adult to the frequency with the child?
I tried to answer this question using:
Freq(adult)/Freq(child)= [(1)/(2pie)] * sqrt (k/(value-for-how-many-times-greater-adult-is-than-child)m)/ [(1)/(2pie)] * sqrt (k/m)
...BUT I got lost because I don't know what to do with the k constants... any ideas??
what is sqrtk/sqrtk? answer: it divides out.
To solve this problem, we can apply the concept of simple harmonic motion and the formula for the frequency of a spring-mass system.
The frequency of oscillation, denoted as f, for a mass-spring system is given by:
f = (1 / (2 * pi)) * sqrt(k / m)
Where:
- f is the frequency of oscillation
- k is the spring constant
- m is the mass of the object
Now, let's consider the child's scenario first:
- The child has a certain mass, let's call it m_child.
- Given the frequency f_child, we can express it as:
f_child = (1 / (2 * pi)) * sqrt(k / m_child)
Next, let's move on to the adult's scenario where the adult's mass is 16 times greater than the child's mass. Therefore, the adult's mass can be expressed as:
m_adult = 16 * m_child
To calculate the frequency of oscillation, f_adult, for the adult, we can substitute m_adult into the frequency formula:
f_adult = (1 / (2 * pi)) * sqrt(k / m_adult)
Now, to find the ratio of the frequency of the adult to the frequency of the child, we can divide f_adult by f_child:
f_adult / f_child = [(1 / (2 * pi)) * sqrt(k / m_adult)] / [(1 / (2 * pi)) * sqrt(k / m_child)]
By simplifying the expression and canceling out the common factors, we get:
f_adult / f_child = sqrt(m_child / m_adult)
Substituting m_adult = 16 * m_child:
f_adult / f_child = sqrt(m_child / (16 * m_child))
f_adult / f_child = sqrt(1 / 16)
f_adult / f_child = 1 / 4
Therefore, the ratio of the frequency of the adult with the mass 16 times greater than the child to the frequency of the child is 1/4.