The compression of 10 cm by a 100 N force on the plane that has a
coefficient of friction of 0.39 give the following values.
- The velocity of the block after the Spring extends 7 cm is approximately 1.73 m/s
- The height at which the block stops rising is approximately 1.1415 m
- The length of the incline is approximately 1.536 m
<h3>How can the velocity and height of the block be calculated?</h3>
Mass of the block, m = 3 kg
![Spring \ constant, K = \dfrac{100 \, N}{0.1 \, m} = \mathbf{ 1000\, N/m}](https://tex.z-dn.net/?f=Spring%20%5C%20constant%2C%20K%20%3D%20%5Cdfrac%7B100%20%5C%2C%20N%7D%7B0.1%20%5C%2C%20m%7D%20%20%3D%20%5Cmathbf%7B%201000%5C%2C%20N%2Fm%7D)
Coefficient of kinetic friction,
= 0.39
Therefore, we have;
Friction force =
·m·g·cos(θ)
Which gives;
Friction force = 0.39 × 3 × 9.81 × cos(48°) ≈ 7.68
Work done by the motion of the block, <em>W</em> ≈ 7.68 × d
The work done = The kinetic energy of the block, which gives;
![\mathbf{\dfrac{1}{2} \times k \cdot x^2 }= 7.68 \cdot d](https://tex.z-dn.net/?f=%5Cmathbf%7B%5Cdfrac%7B1%7D%7B2%7D%20%5Ctimes%20k%20%5Ccdot%20x%5E2%20%7D%3D%207.68%20%5Ccdot%20d)
The initial kinetic energy in the spring is found as follows;
K.E. = 0.5 × 1000 N/m × (0.1 m)² = 5 J
The initial velocity of the block is therefore;
5 = 0.5·m·v²
v₁ = √(2 × 5 ÷ 3) ≈ 1.83
Work done by the motion of the block, <em>W</em> ≈ 7.68 N × 0.07 m ≈ 0.5376 J
Chane in kinetic energy, ΔK.E. = Work done
ΔK.E. = 0.5 × 3 × (v₁² - v₂²)
Which gives;
ΔK.E. = 0.5 × 3 × (1.83² - v₂²) = 0.5376
Which gives;
- The velocity of the block after the Spring extends 7 cm, v₂ ≈ <u>1.73 m/s</u>
The height at which the block will stop moving, <em>h</em>, is given as follows;
![At \ the \ maximum \ height, \ h, \ we \ have ; \ \dfrac{1}{2} \times 1000 \times 0.1^2 = 7.68 \times x](https://tex.z-dn.net/?f=At%20%5C%20the%20%5C%20maximum%20%5C%20height%2C%20%5C%20h%2C%20%5C%20we%20%5C%20have%20%3B%20%5C%20%20%5Cdfrac%7B1%7D%7B2%7D%20%5Ctimes%201000%20%5Ctimes%200.1%5E2%20%3D%207.68%20%5Ctimes%20x)
Which gives;
![Length \ of \ the \ incline \ at \ maximum \ height, \ x_{max} =\dfrac{ 7.68 }{ \dfrac{1}{2} \times 1000 \times 0.1^2 } \approx 1.536](https://tex.z-dn.net/?f=Length%20%5C%20of%20%5C%20the%20%5C%20incline%20%5C%20at%20%5C%20maximum%20%5C%20height%2C%20%5C%20x_%7Bmax%7D%20%3D%5Cdfrac%7B%20%207.68%20%7D%7B%20%5Cdfrac%7B1%7D%7B2%7D%20%5Ctimes%201000%20%5Ctimes%200.1%5E2%20%20%7D%20%5Capprox%201.536)
The distance up the inclined, the block rises, at maximum height is therefore;
≈ 1.536 m
Therefore;
h = 1.536 × sin(48°) ≈ 1.1415
- The height at which the block stops rising, h ≈ <u>1.1415 m</u>
From the above solution for the height, the length of the incline is he
distance along the incline at maximum height which is therefore;
- Length of the incline,
= 1.536 m
Learn more about conservation of energy here:
brainly.com/question/7538238