A 3.00-kg mass is fastened to a light spring that passes over a pulley. They pulley is frictionless, and its inertia may be neglected. The mass is released from rest when the spring is unstretched. If the mass drops 10.0cm before stopping, find (a) the spring constant of the spring and (b) the speed of the mass when it is 5.00cm below its starting point.

When the spring stops, there is no kinetic energy and the potential energy lost (M g X) equals the added potential energy of the spring, (1/2) k X^2.
(a) Since you know M = 3.00 kg, g = 9.8 m/s^2 and H = 0.100 mm, you can solve for the spring constant k.
(b) Use the enery conservation law and the fact that the energy stored in the spring is 1/4 as large when X = 5 cm, as it is at X = 10 cm.

(a) Well, let's solve for the spring constant, shall we?

We know that the potential energy lost is equal to the added potential energy of the spring. So, we can write the equation as:
M g X = (1/2) k X^2

Now plug in the values:
(3.00 kg)(9.8 m/s^2)(0.10 m) = (1/2) k (0.10 m)^2

Simplifying, we get:
2.94 J = (1/200) k

Now, using some math magic, we can solve for the spring constant, k:
k = 200 * 2.94 J

Calculating that out, we get:
k ≈ 588 N/m

So, the spring constant is approximately 588 N/m.

(b) Now, let's use the energy conservation law to figure out the speed of the mass when it is 5.00 cm below its starting point.

We know that the energy stored in the spring is 1/4 as large when X = 5 cm, as it is at X = 10 cm.

So, we can write the equation:
(1/4) k (0.05 m)^2 = (1/2) M v^2

Now plug in the values:
(1/4) (588 N/m) (0.05 m)^2 = (1/2) (3.00 kg) v^2

Solving for the speed, v:
v^2 = (1/4) (588 N/m) (0.05 m)^2 / (1/2) (3.00 kg)

v^2 ≈ (147 N/m) (0.0025 m^2) / 1.5 kg

v^2 ≈ 0.0245 m^2/s^2

Taking the square root, we get:
v ≈ 0.1565 m/s

So, the speed of the mass when it is 5.00 cm below its starting point is approximately 0.1565 m/s.

(a) To solve for the spring constant k, we can use the equation for potential energy:

Potential energy lost = Potential energy stored in the spring

M * g * X = (1/2) * k * X^2

Given:
M = 3.00 kg
g = 9.8 m/s^2
X = 0.100 m

Substituting the values into the equation, we get:

3.00 kg * 9.8 m/s^2 * 0.100 m = (1/2) * k * (0.100 m)^2

2.94 J = (1/2) * k * 0.010 m^2

Multiplying both sides by 2 and dividing by 0.010 m^2, we get:

k = 2.94 J / (0.010 m^2 * 2)
k = 147 N/m

Therefore, the spring constant of the spring is 147 N/m.

(b) To find the speed of the mass when it is 5.00 cm below its starting point, we can use the conservation of energy:

Initial potential energy + initial kinetic energy = Final potential energy + Final kinetic energy

Since the mass is initially at rest, the initial kinetic energy is zero. Also, when the mass is 5.00 cm below the starting point, the energy stored in the spring is 1/4 of the energy stored when the mass is 10.0 cm below the starting point.

Let's denote the height when the mass is 10.0 cm below the starting point as X1 and the height when the mass is 5.00 cm below the starting point as X2.

Therefore, the final potential energy is (1/4) times the initial potential energy:

(1/4) * (1/2) * k * X1^2 = (1/2) * k * X2^2

Substituting the values, we have:

(1/4) * (1/2) * 147 N/m * (0.100 m)^2 = (1/2) * 147 N/m * (0.050 m)^2

0.018375 J = 0.03675 J

Since the initial kinetic energy is zero, the final kinetic energy is equal to the remaining energy:

Final kinetic energy = Final potential energy

(1/2) * M * v^2 = 0.018375 J

Substituting the value for M, we can solve for v:

(1/2) * 3.00 kg * v^2 = 0.018375 J

1.5 * v^2 = 0.018375 J

v^2 = 0.018375 J / 1.5

v^2 = 0.01225 J

Taking the square root of both sides, we get:

v ≈ 0.1108 m/s

Therefore, when the mass is 5.00 cm below its starting point, the speed of the mass is approximately 0.1108 m/s.

To solve this problem, we need to use the principles of energy conservation. We know that the potential energy lost by the mass as it drops is equal to the potential energy gained by the spring. Let's go through the steps to find the spring constant and the speed of the mass.

(a) To find the spring constant, we need to equate the potential energy lost and the potential energy gained by the spring.

Potential Energy Lost = Potential Energy Gained
M*g*X = (1/2)*k*X^2

Substituting the given values, we have:
3.00 kg * 9.8 m/s^2 * 0.10 m = (1/2) * k * (0.10 m)^2

Now we can solve for k:
k = (3.00 kg * 9.8 m/s^2 * 0.10 m) / ((1/2) * (0.10 m)^2)

Calculating this expression, we get:
k ≈ 588 N/m

Therefore, the spring constant is approximately 588 N/m.

(b) To find the speed of the mass when it is 5.00 cm below its starting point, we can use the principle of energy conservation.

When the mass is at 10 cm below its starting point, the energy stored in the spring is given by:
(1/2) * k * (0.10 m)^2

When the mass is at 5 cm below its starting point, the energy stored in the spring is given by 1/4 of the energy at 10 cm below its starting point. Therefore, the energy stored in the spring is:
(1/4) * (1/2) * k * (0.10 m)^2

Now, we can equate the initial potential energy (at 10 cm below) to the final potential energy (at 5 cm below) and the kinetic energy of the mass.

Potential Energy (initial) + Kinetic Energy (initial) = Potential Energy (final) + Kinetic Energy (final)

(1/2) * k * (0.10 m)^2 = (1/4) * (1/2) * k * (0.10 m)^2 + (1/2) * M * V^2

Substituting the given values, we have:
(1/2) * (588 N/m) * (0.10 m)^2 = (1/4) * (1/2) * (588 N/m) * (0.10 m)^2 + (1/2) * 3.00 kg * V^2

Simplifying this equation, we can solve for the speed V:
(1/2) * (588 N/m) * (0.10 m)^2 - (1/4) * (1/2) * (588 N/m) * (0.10 m)^2 = (1/2) * 3.00 kg * V^2

Calculating this expression, we find:
V ≈ 0.692 m/s

Therefore, the speed of the mass when it is 5.00 cm below its starting point is approximately 0.692 m/s.