Answer :
To find the compression length of the eraser, resolve the force, calculate stress, use Hooke's law for strain, and then compute the change in length.
To determine the compression length of the pencil eraser when a vertical force of 6.00 N is applied at an angle of 20.0 degrees to the horizontal, several steps are involved:
1. Resolve the Force Components: First, resolve the vertical force (F) into its vertical (\(F_v\)) and horizontal ((F_h) components using trigonometry. (F_v = 6.00 N * sin(20.0) and (F_h = 6.00 N * \cos(20.0).
2. Calculate the Cross-Sectional Area (A): Determine the cross-sectional area (A) of the pencil eraser, which is needed to calculate stress. The pencil's diameter is 6.00 mm, so the radius (r) is 3.00 mm (0.003 m). Use the formula (A = pi * r^2) to find (A).
3. Calculate Stress (sigma): Calculate the stress (sigma) applied to the eraser using the formula (sigma = {F_v} / {A}), where (F_v) is the vertical force component.
4. Determine Strain ( epsilon ): Calculate the strain (epsilon) using Hooke's law, which relates stress to strain: (epsilon = sigma / Y ), where (Y) is the Young's modulus for the eraser material. Typically, for rubber-like materials, (Y) is in the range of 0.01 to 0.1 GPa.
5. Calculate Compression Length (Delta L): Finally, calculate the change in length (Delta L) of the eraser using the formula (Delta L = epsilon * L), where (epsilon) is the strain, and (L) is the original length of the eraser.
By following these steps and plugging in the values, you can determine how much the pencil eraser is compressed lengthwise under the given force and angle conditions. This calculation is essential in understanding the deformation of materials under external forces and can be applied to various engineering and material science scenarios.
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