The position of a 2.75×10^5 N training helicopter under test is given by r =(0.020m/s 3 )t^3 i ^ +(2.2m/s)tj ^ −(0.060m/s 2 )t^2 k ^. find the net force of the helicopter at t=5.0s
the actual answer for two sig figs are 1.7x10^4*i^-3.3x10^3*k^1
Well, the net force of the helicopter can be found by taking the derivative of the position function with respect to time. Let's get into some calculations!
Given: r = (0.020m/s^3)t^3 i^ + (2.2m/s)t j^ - (0.060m/s^2)t^2 k^
Taking the derivative of each component, we have:
dr/dt = (0.060m/s^3)t^2 i^ + (2.2m/s) j^ - (0.120m/s^2)t k^
Now, to find the net force, we multiply the mass of the helicopter by the acceleration:
F = m * a
Since the mass is not given, we won't be able to calculate the actual net force. But hey, if we were to make an educated guess, we could say the net force is powerful enough to make the helicopter fly high into the sky! Just like a superhero helicopter!
Remember, always take calculations with a pinch of humor.
To find the net force of the helicopter at t=5.0s, we need to differentiate the position vector with respect to time twice, which will give us the acceleration vector. Then, we can multiply the acceleration vector by the mass of the helicopter to calculate the net force.
Let's start by differentiating the position vector with respect to time to find the velocity vector:
v = dr/dt
Differentiating r =(0.020m/s^3)t^3 i^ +(2.2m/s)t j^ -(0.060m/s^2)t^2 k^ with respect to t, we get:
v = (0.060m/s^3)t^2 i^ + (2.2m/s) j^ - (0.120m/s^2)t k^
Next, we differentiate the velocity vector with respect to time to find the acceleration vector:
a = dv/dt
Differentiating v = (0.060m/s^3)t^2 i^ + (2.2m/s) j^ - (0.120m/s^2)t k^ with respect to t, we get:
a = (0.120m/s^3)t i^ - 0.120m/s^2 k^
Now that we have the acceleration vector, we can find the net force by multiplying it by the mass of the helicopter. However, we need to know the mass of the helicopter in order to calculate the net force accurately. Once we have the mass value, we can substitute t=5.0s into the acceleration vector and multiply it by the mass to obtain the net force.
r =(0.020m/s³)t ³ i ^ +(2.2m/s)tj ^ −(0.060m/s² )t²k ^.
v= dr/dt =(3•0.020m/s³ ) •t²• i ^ +(2.2m/s) • j ^ −2• (0.060m/s² )t•k ^.
a=d²r/dt²=(6•0.020m/s ³) •t i ^ -2• (0.060m/s² )•k ^.
F=ma =2.75•10⁵•{(6•0.020m/s ³) •t i ^ -2• (0.060m/s² )•k ^.}
t=5 s
F₅=2.75•10⁵•{(6•0.020m/s ³) •5• i ^ -2• (0.060m/s² )•k ^.}=
=165000 i ^ - 33000•k ^.