A) The total energy of the system is sum of kinetic energy and elastic potential energy:
![E=K+U= \frac{1}{2}mv^2 + \frac{1}{2}kx^2](https://tex.z-dn.net/?f=E%3DK%2BU%3D%20%5Cfrac%7B1%7D%7B2%7Dmv%5E2%20%2B%20%20%5Cfrac%7B1%7D%7B2%7Dkx%5E2%20%20)
where
m is the mass
v is the speed
k is the spring constant
x is the elongation/compression of the spring
The total energy is conserved, so we can calculate its value at any point of the motion. If we take the point of maximum displacement:
![x=A=4.00 cm = 0.04 m](https://tex.z-dn.net/?f=x%3DA%3D4.00%20cm%20%3D%200.04%20m)
then the velocity of the system is zero, so the total energy is just potential energy, and it is equal to
![E=U= \frac{1}{2}kA^2 = \frac{1}{2}(35.0 N/m)(0.04 m)^2=0.028 J](https://tex.z-dn.net/?f=E%3DU%3D%20%5Cfrac%7B1%7D%7B2%7DkA%5E2%20%3D%20%20%5Cfrac%7B1%7D%7B2%7D%2835.0%20N%2Fm%29%280.04%20m%29%5E2%3D0.028%20J%20%20)
b) When the position of the object is
![x=1.00 cm = 0.01 m](https://tex.z-dn.net/?f=x%3D1.00%20cm%20%3D%200.01%20m)
the potential energy of the system is
![U= \frac{1}{2}kx^2 = \frac{1}{2}(35.0 N/m)(0.01 m)^2 = 1.75 \cdot 10^{-3} J](https://tex.z-dn.net/?f=U%3D%20%5Cfrac%7B1%7D%7B2%7Dkx%5E2%20%3D%20%20%5Cfrac%7B1%7D%7B2%7D%2835.0%20N%2Fm%29%280.01%20m%29%5E2%20%3D%201.75%20%5Ccdot%2010%5E%7B-3%7D%20J%20%20)
and so the kinetic energy is
![K=E-U=0.028 J - 1.75 \cdot 10^{-3}J =0.026 J](https://tex.z-dn.net/?f=K%3DE-U%3D0.028%20J%20-%201.75%20%5Ccdot%2010%5E%7B-3%7DJ%20%3D0.026%20J)
since the mass is
![m=50.0 g=0.05 kg](https://tex.z-dn.net/?f=m%3D50.0%20g%3D0.05%20kg)
, and the kinetic energy is given by
![K= \frac{1}{2}mv^2](https://tex.z-dn.net/?f=K%3D%20%5Cfrac%7B1%7D%7B2%7Dmv%5E2%20)
we can re-arrange the formula to find the speed of the object:
![v= \sqrt{ \frac{2K}{m} }= \sqrt{ \frac{2 \cdot 0.026 J}{0.05 kg} }=1.02 m/s](https://tex.z-dn.net/?f=v%3D%20%5Csqrt%7B%20%5Cfrac%7B2K%7D%7Bm%7D%20%7D%3D%20%5Csqrt%7B%20%5Cfrac%7B2%20%5Ccdot%200.026%20J%7D%7B0.05%20kg%7D%20%7D%3D1.02%20m%2Fs%20%20)
c) The potential energy when the object is at
![x=3.00 cm=0.03 m](https://tex.z-dn.net/?f=x%3D3.00%20cm%3D0.03%20m)
is
![U= \frac{1}{2}kx^2 = \frac{1}{2}(35.0 N/m)(0.03 m)^2 =0.016 J](https://tex.z-dn.net/?f=U%3D%20%5Cfrac%7B1%7D%7B2%7Dkx%5E2%20%3D%20%20%5Cfrac%7B1%7D%7B2%7D%2835.0%20N%2Fm%29%280.03%20m%29%5E2%20%3D0.016%20J%20%20)
Therefore the kinetic energy is
![K=E-U=0.028 J-0.016 J = 0.012 J](https://tex.z-dn.net/?f=K%3DE-U%3D0.028%20J-0.016%20J%20%3D%200.012%20J)
d) We already found the potential energy at point c, and it is given by