Respuesta :
The net force on the helicopter at t=5.0s = F = 1.65.10⁴i - 3.3.10³k
Further explanation
Newton's 2nd law explains that the acceleration produced by the resultant force on an object is proportional and in line with the resultant force and inversely proportional to the mass of the object
[tex]\large{\boxed{\bold{a~=~\frac{F}{m} }}}[/tex]
∑F = m. a
F = force, N
m = mass = kg
a = acceleration due to gravity, m / s²
While the speed is the first derivative of distance
[tex]\large{\boxed{\bold{V~=~\frac{dr}{dt} }}}[/tex]
While acceleration is the first derivative of the velocity function for t
[tex]\large{\boxed{\bold{a~=~\frac{dv}{dt} }}}[/tex]
It is known that the distance function (r) = (0.020 t³i + 2.2 tj-0.06t²k
Then the speed (v)
[tex]v~=~\frac{dr}{dt}[/tex]
v = 0.06t²i + 2.2j - 0.12tk
Then acceleration (a)
[tex]a~=~\frac{dv}{dt}[/tex]
a = 0.12ti - 0.12k
While :
W = m . g
[tex]m~=~\frac{W}{g}[/tex]
m = 2.75.10⁵ : 10
m = 2.75.10⁴
so net force :
F = m . a
F = 2.75.10⁴ . ( 0.12ti - 0.12k)
for t = 5
F = 2.75.10⁴ . ( 0.12.5i - 0.12k)
F = 2.75.10⁴ (0.6i-0.12k)
F = 1.65.10⁴i - 3.3.10³k
Learn more
Newton's second law
https://brainly.com/question/10673278
the acceleration of an object
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https://brainly.com/question/1998887
a correct statement of Newton's second law
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Keywords : Newton's Law, acceleration, net force
The net force on the helicopter at time [tex]t = 5.0\,{\text{s}}[/tex] is [tex]\fbox{\begin\\\left( {1.68 \times {{10}^4}\,\hat i - 3.36 \times {{10}^3}\,\hat j} \right)\,{\text{N}}\end{minispace}}[/tex].
Further Explanation:
The helicopter moves under the action of the force which changes the position of the helicopter with respect to time according to the following equation.
[tex]\vec{r}=(0.020\text{ m}/\text{s}^3)\text{ t}^3\hat{i}+(2.2\text{ m}/\text{s})\text{ t}^3\hat{j}-(0.060\text{ m}/\text{s}^2)\text{ t}^2\hat{k}[/tex]
The force acting on the helicopter at a particular time is given by:
[tex]\fbox{\begin\\F= m\cdot{a}\end{minispace}}[/tex]
Here, [tex]m[/tex] is the mass of the helicopter and [tex]a[/tex] is the acceleration of helicopter.
The helicopter has a weight of [tex]2.75 \times {10^5}\,{\text{N}}[/tex]. This weight of the helicopter can be converted into the mass of the helicopter.
The weight of the helicopter can be represented as.
[tex]\fbox{\begin\\W=m \cdot g\end{minispace}}[/tex]
Rearrange and simplify the expression for the mass of helicopter.
[tex]\begin{aligned}m&=\frac{W}{g}\\&=\frac{{2.75\times{{10}^5}}}{{9.8}}\,{\text{kg}}\\&=2.{\text{80}}\times{\text{1}}{{\text{0}}^4}\,{\text{kg}}\\\end{aligned}[/tex]
The velocity of the helicopter can be determined by the differentiation of the position vector of the helicopter with respect to time.
[tex]\begin{aligned}\vec{v}&=\dfrac{{d\vec{r}}}{{dt}}\\&=\dfrac{d}{{dt}}\ [(0.020\text{ m}/\text{s}^3)\text{ t}^3\hat{i}+(2.2\text{ m}/\text{s})\text{ t}^3\hat{j}-(0.060\text{ m}/\text{s}^2)\text{ t}^2\hat{k}]\\&=0.060{t^2}\hat{i}+2.2\hat{j}-0.12t\hat{k}\end{aligned}[/tex]
The acceleration of the helicopter is the derivative of the velocity with respect to the time.
[tex]\begin{aligned}\vec a&=\frac{{d\vec v}}{{dt}}\\&=\frac{d}{{dt}}\left({0.060{t^2}\hat i+2.2\hat j-0.12t\hat k}\right)\\&=0.12t\hat i-0.12\hat k\\\end{aligned}[/tex]
Substitute the value of [tex]t = 5.0\,{\text{s}}[/tex] in equation of the acceleration.
[tex]\begin{aligned}a&=0.12\left({5.0}\right)\hat i-0.12\hat k\\&=0.6\hat i-0.12\hat k\\\end{aligned}[/tex]
Substitute the values of the acceleration and the mass of the helicopter in the equation of force.
[tex]\begin{aligned}F&=\left( {2.80\times{{10}^4}\,{\text{kg}}}\right)\left({0.6\hat i-0.12\hat k}\right) \\&=\left({1.68\times{{10}^4}\,\hat i-3.36\times{{10}^3}\,\hat j}\right)\,{\text{N}}\\\end{aligned}[/tex]
Thus, the net force acting on the helicopter at time [tex]t = 5\,{\text{s}}[/tex] is [tex]\fbox{\begin\\\left( {1.68 \times {{10}^4}\,\hat i - 3.36 \times {{10}^3}\,\hat j} \right)\,{\text{N}}\end{minispace}}[/tex].
Learn More:
1. The motion of a body under friction https://brainly.com/question/7031524
2. Conservation of energy https://brainly.com/question/3943029
3. Motion in two dimensions https://brainly.com/question/11023695
Answer Details:
Grade: College
Subject: Physics
Chapter: Force and acceleration
Keywords:
Helicopter, force, time, 5s, position, velocity, derivative, acceleration, weight, net force, t=5s, 0.020t3 m/s3, 2.75x10^5 N, helicopter under test, gravity, mass of helicopter.
