Question

In the future, parents will no longer need to push their child on a swing. You have invented an automated child launching device. Your device is modeled as shown. The length of the swing's rope is 3 m and your child has a mass of 40 kg. What spring constant will you require if it can only compress 30 cm before it will launch your loved one to a maximum 45 degree angle? what max acceleration will the kid experience after leaving the spring? what avg acceleration will the child experience?

Answer 1

Data:

$\theta =45^{\circ}$

Part a)

Applying the conservation of energy we have:

$E=0$

or

$E_{i}=E_{f}$

$U_{ei}=U_{g}+K_{f}$

The speed of the kid at the highest point (45 degrees) is zero, so; $K_{f}=0$

$U_{ei}=U_{g}$

$\frac{1}{2}kx^{2}=mgH$

Isolating :

$k=\frac{2mgH}{x^{2}}\, \, \, \, \, \, \, \, \, \, (1)$

Where, is the max height reaches for the kid, and is calulate using:

$H =l-l\cos 45=3-3\cos 45$

Replacing values given in the statement of the problem and the value of H in equation (1) :

$k=\frac{2mgH}{x^{2}}\, \, \, \, \, \, \, \, \, \, (1)$

$k=\frac{2(40\, kg)(9.8\, m/s^{2})(0.88\, m)}{(0.3\, m)^{2}}$