Question

a 3 kg mass is hanging from a spring. the spring has a coefficient of 400 n/m and is stretched 0.5 m. the mass is 2 m above the ground and moving at 4 m/s.

1. how much gravitational potential energy does the mass have?
2. how much spring potential energy does the mass have?
3. how much kinetic energy does the mass have?
4. how much total energy does the mass have?

Explanation:

Step1: Calculate gravitational potential energy

Use formula $U_g = mgh$, where $m=3\ \text{kg}$, $g=9.8\ \text{m/s}^2$, $h=2\ \text{m}$.
$U_g = 3 \times 9.8 \times 2$

Step2: Calculate spring potential energy

Use formula $U_s = \frac{1}{2}kx^2$, where $k=400\ \text{N/m}$, $x=0.5\ \text{m}$.
$U_s = \frac{1}{2} \times 400 \times (0.5)^2$

Step3: Calculate kinetic energy

Use formula $K = \frac{1}{2}mv^2$, where $m=3\ \text{kg}$, $v=4\ \text{m/s}$.
$K = \frac{1}{2} \times 3 \times (4)^2$

Step4: Calculate total energy

Sum gravitational, spring, and kinetic energy.
$E_{total} = U_g + U_s + K$

Answer:

1. Gravitational potential energy: $58.8\ \text{J}$
2. Spring potential energy: $50\ \text{J}$
3. Kinetic energy: $24\ \text{J}$
4. Total energy: $132.8\ \text{J}$