Answer:
-30 m/s²
Explanation:
The total energy is the sum of the gravitational potential energy of the mass, the kinetic energy of the mass, and the elastic energy in the spring.
∑E = PE + KE + EE
∑E = mgh + ½ mv² + ½ kx²
At the bottom, h = 0 and v = 0.
∑E = ½ kx²
2 = ½ (400) x²
2 = 200 x²
0.01 = x²
x = 0.1
At the top, v = 0 and x = 0.1 − h.
∑E = mgh + ½ kx²
2 = (1) (10) h + ½ (400) (0.1 − h)²
2 = 10h + 200 (0.01 − 0.2h + h²)
2 = 10h + 2 − 40h + 200h²
0 = 200h² − 30h
0 = 10h (20h − 3)
h = 0.15
So x ranges from -0.05 m to 0.10 m.
Sum of forces on the mass:
∑F = ma
kx − mg = ma
The minimum acceleration is when x is a minimum.
(400 N/m) (-0.05 m) − (1 kg) (10 m/s²) = (1 kg) a
a = -30 m/s²
The maximum acceleration is when x is a maximum.
(400 N/m) (0.1 m) − (1 kg) (10 m/s²) = (1 kg) a
a = 30 m/s²
And of course, acceleration is 0 when kx = mg.
(400 N/m) x = (1 kg) (10 m/s²)
x = 0.025 m
