Q48E - University Physics with Modern Physics

Question

An ingenious bricklayer builds a device for shooting bricks up to the top of the wall where he is working. He places a brick on a vertical compressed spring with force constant $\mathit{k}\mathbf{=}\mathbf{450}\mathbf{ }\mathbf{\text{N/m}}$and negligible mass. When the spring is released, the brick is propelled upward. If the brick has mass $\mathbf{1}.\mathbf{80} \mathbf{kg}$ and is to reach a maximum height of $\mathbf{3}.\mathbf{6} \mathbf{m}$ above its initial position on the compressed spring, what distance must the bricklayer compress the spring initially? (The brick loses contact with the spring when the spring returns to its uncompressed length. Why?)

Step-by-Step Solution

Answer

$0.531 \text{m}$

Step 1: Identification of given data

Mass of brick is $m=1.8 \mathrm{kg}$

Spring constant is $\mathrm{k}=450 \text{N/m}$

Maximum height is $h=3.6 \mathrm{m}$

Step 2: Significance of law of conservation of energy

Energy cannot be created or destroyed; it can only be transformed from one form into another or transferred from one object to another, according to the physical rule of conservation of energy.

When an object is compressed or under strain, such as when a rubber band is stretched, elastic potential energy is stored (when you squeeze a spring). The kinetic energy, also known as the energy of motion, is created when the potential energy is "released."

$\text{Elastic} \text{Potentia} \text{Energy}\left(\frac{1}{2}k{x}^2\right)=\text{Potential} \text{Energy}\left(mgh\right)$                                                 ...(i)

Where, $k$ is spring constant, $x$ is the initial compression, $m$ is the mass, $g$ is the acceleration due to gravity and $h$ is the height

Step 3: Determining the distance must the bricklayer compress the spring initially

Using the equation (i)

$\frac{1}{2}k{x}^2=mgh$

Substitute all the values in equation (i)

$\begin{array}{rcl}x& =& \sqrt{\frac{2\times 1.8 \text{kg}\times 9.8 {\text{m/s}}^2\times 3.6 \text{m}}{450 \text{N/m}}}\\ & =& 0.531 \text{m}\end{array}$

Hence the distance must the bricklayer compresses the spring initially is$0.531 \text{m}$

The brick loses contact with the spring when the spring returns to its uncompressed length because at the uncompressed length of the spring the speed is maximum. After crossing this position, the speed gradually decreases. The spring losses the potential energy that would be gained by the brick in the form of kinetic energy. When bricks gain kinetic energy it lose contact with spring.