A 100-g block hangs from a spring with k = 5.2 N/m. At t = 0 s, the block is 20.0 cm below the equilibrium position and moving upward with a speed of 216 cm/s. What is the block's speed when the displacement from equilibrium is 27.0 cm?

The answer is 172 cm/s; but I cannot figure out how to get this answer.

You need to consider gravitational potential energy, spring potential energy, and block kinetic energy. The sum of the three is constant. I will refer gravitational P.E. to the equilibrium position

(-0.2 m) M*g + (1/2)k(0.2)^2 + (M/2)(2.16 m/s)^2
= (-0.27 m)M*g + (1/2)k(0.27)^2 + (M/2)V^2

(M/2)V^2 = 0.0686 J + 0.2333 J - 0.0855 J = 0.2164 J

V^2 = 4.328 m^2/s^2
V = 2.08 m/s

In order to get the 1.72 m/s answer, you would have to negelect the gravitational potential energy term. I don't agree with doing that.

When I follow your steps I get 1284 cm/s, so I'm not sure how you're getting 2.08 m/s.

What gravitational potential energy term are you referring to?

To find the block's speed when the displacement from equilibrium is 27.0 cm, we can use the principles of simple harmonic motion.

First, let's determine the spring constant k in SI units. We know that k = 5.2 N/m, so k = 5.2 N/m * (1 N / 1 kg*m/s^2) = 5.2 kg/s^2.

Next, let's determine the amplitude of the oscillation. The amplitude of the oscillation is the maximum displacement from equilibrium. In this case, the block is 20.0 cm below the equilibrium position, so the amplitude is 20.0 cm = 0.20 m.

To find the block's speed when the displacement is 27.0 cm, we can use the equation for the speed of an object undergoing simple harmonic motion:

v = ω * A * √(1 - (x / A)^2),

where v is the speed, ω is the angular frequency, A is the amplitude, and x is the displacement from equilibrium.

First, let's find the angular frequency ω using the formula:

ω = √(k / m),

where m is the mass of the block. We know that the block has a mass of 100 g = 0.100 kg.

ω = √(5.2 kg/s^2 / 0.100 kg) = 7.211 s^-1.

Now, we can substitute the values of ω, A, and x into the equation for speed:

v = (7.211 s^-1) * (0.20 m) * √(1 - (0.27 m / 0.20 m)^2) = 6.20 m/s = 620 cm/s.

Therefore, the block's speed when the displacement from equilibrium is 27.0 cm is 620 cm/s.

It seems that there may be a mistake in the calculation. Could you please check the problem statement or double-check your calculations?