1.) A vertical spring with a spring constant of 450 N/m is mounted on the floor. From directly above the spring, which is unstrained, a 0.30 kg block is dropped from rest. It collides with and sticks to the spring, which is compressed 2.5 cm in bringing the block to a momentary halt. Air resistance neglible, from what height above the compressed string was the block dropped?

(Steps please?

Oh boy, let me juggle these numbers for you! First, we need to figure out the potential energy the block has when it is dropped. We know that potential energy (PE) is given by the equation PE = m*g*h, where m is the mass of the block (0.30 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height above the spring.

Now, when the block hits the spring, all of that potential energy is converted into the elastic potential energy of the compressed spring. The formula for elastic potential energy (PEelastic) is PEelastic = (1/2) * k * x^2, where k is the spring constant (450 N/m), and x is the amount of compression (2.5 cm or 0.025 m).

Since the block comes to a momentary halt, the elastic potential energy must be equal to the initial potential energy, so we have:

PE = PEelastic

m*g*h = (1/2) * k * x^2

Now let's solve for h! We have:

h = [(1/2) * k * x^2] / (m * g)

Plugging in the values, we get:

h = [(1/2) * 450 N/m * (0.025 m)^2] / (0.30 kg * 9.8 m/s^2)

h = [0.28125 N * m] / [2.94 N]

h ≈ 0.096 m

So, the block was dropped from a height of approximately 0.096 meters above the compressed spring. I hope that calculation didn't make your head spin too much!

To solve this problem, we can apply the principle of conservation of mechanical energy. The initial potential energy of the block when it is dropped gets converted into the spring potential energy when the block is compressed.

Here's how you can solve this problem step by step:

Step 1: Identify the given information:
- Spring constant: k = 450 N/m
- Mass of the block: m = 0.30 kg
- Compression of the spring: x = 2.5 cm = 0.025 m

Step 2: Determine the potential energy of the block when it was dropped:
The potential energy of the block when it was dropped is given by the formula PE = mgh, where m is the mass, g is the acceleration due to gravity (9.8 m/s^2), and h is the height above the compressed spring.

Step 3: Determine the potential energy of the spring when it is compressed:
The potential energy of the compressed spring can be calculated using the formula PE = (1/2)kx^2, where k is the spring constant and x is the compression of the spring.

Step 4: Equate the potential energies:
Since mechanical energy is conserved, we can equate the potential energy of the block when it was dropped with the potential energy of the spring when it is compressed. This gives us the equation mgh = (1/2)kx^2.

Step 5: Solve for h:
Rearrange the equation to solve for h: h = (1/2)(kx^2)/(mg).

Step 6: Substitute the given values:
Plug in the values for k, x, m, and g into the equation and calculate the height h.

Step 7: Calculate the answer:
Calculate the height h using the given values in Step 6. The resulting value will be the height above the compressed spring from which the block was dropped.

Following these steps, you should be able to calculate the height above the compressed spring from which the block was dropped.

The height from which the block was dropped, which is H above the compressed spring, satisfies the condition

gravitional P.E. lost = spring P.E. gained
M g H = (1/2) k X^2

They tell you what k, M and X are. Solve for H.

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