Askiitians Tutor Team
Last Activity: 16 Days ago
To determine the maximum compression of a spring when a 1 kg block hits it, we can use the principles of energy conservation. When the block falls and hits the spring, its kinetic energy is converted into potential energy stored in the spring. Let's break this down step by step.
Understanding the Energy Transfer
When the block is in free fall, it possesses gravitational potential energy, which is converted into kinetic energy as it falls. Upon hitting the spring, this kinetic energy is transferred to the spring, compressing it. The maximum compression occurs when all the kinetic energy has been converted into elastic potential energy in the spring.
Key Formulas
We will use the following formulas:
- Kinetic Energy (KE): KE = (1/2)mv²
- Potential Energy in Spring (PE): PE = (1/2)kx²
Here, m is the mass of the block, v is its velocity just before hitting the spring, k is the spring constant, and x is the maximum compression of the spring.
Calculating the Velocity of the Block
First, we need to find the velocity of the block just before it hits the spring. If we assume the block falls from a height h, we can use the formula for gravitational potential energy:
PE = mgh
At the point of impact, this potential energy converts to kinetic energy:
mgh = (1/2)mv²
We can simplify this to find the velocity:
v² = 2gh
Thus, v = √(2gh).
Setting Up the Energy Conservation Equation
Now, we can set the kinetic energy equal to the potential energy stored in the spring at maximum compression:
(1/2)mv² = (1/2)kx²
Substituting for v, we get:
(1/2)m(2gh) = (1/2)kx²
This simplifies to:
mgh = (1/2)kx²
Solving for Maximum Compression
Rearranging the equation to solve for x gives:
x² = (2mgh) / k
Taking the square root of both sides, we find:
x = √((2mgh) / k).
Example Calculation
Let’s say the block falls from a height of 2 meters and the spring constant k is 1000 N/m. Plugging in the values:
- m = 1 kg
- g = 9.81 m/s²
- h = 2 m
- k = 1000 N/m
Now substituting these into the equation:
x = √((2 * 1 kg * 9.81 m/s² * 2 m) / 1000 N/m)
x = √((39.24) / 1000)
x = √0.03924 ≈ 0.198 m or 19.8 cm.
Final Thoughts
This means that the maximum compression of the spring, when a 1 kg block falls from a height of 2 meters and hits a spring with a spring constant of 1000 N/m, would be approximately 19.8 cm. This approach illustrates how energy conservation principles can be applied to solve real-world physics problems effectively.