Respuesta :
Answer:
Velocity =0.241 m/s
Acceleration = 7.21e-4 m/s²
Explanation:
The wheel travels through
Θ = (7.40/37.3)*360º = 71.42º
and so the length of the line segment connecting the initial and final position is
L = 2*L*sin(Θ/2) = 2 * (183m/2) * sin(71.42º/2) = 107 m
so the average velocity is
v = L / t = 107m / 7.40*60s = 0.241 m/s
Initially, let's say the velocity is along the +x axis:
Vi = π * 183m / (37.3*60s) i = 0.257 m/s i
Later, it's rotated through 71.42º, so
Vf = 0.257m/s * (cos71.42º i + sin71.42º j) = [0.0819 i + 0.244 j] m/s
ΔV = Vf - Vi = [(0.0819 - 0.257) i + 0.244 j] m/s = [-0.175 i + 0.244 j] m/s
which has magnitude
|ΔV| = √(0.175² + 0.244²) m/s = 0.300 m/s
Then the average acceleration is
a_avg = |ΔV| / t = 0.300m/s / (7.40*60s) = 6.76e-4 m/s²
The instantaneous acceleration is centripetal: a = ω²r
a = (2π rads / (37.3*60s)² * 183m/2 = 7.21e-4 m/s²
Answer:
[tex]v = 0.24 m/s[/tex]
[tex]a = 6.75 \times 10^{-4} m/s^2[/tex]
Explanation:
Given that wheel completes one round in total time T = 37.3 min
so angular speed of the wheel is given as
[tex]\omega = \frac{2\pi}{T}[/tex]
[tex]\omega = \frac{2\pi}{37.3} rad/min[/tex]
now the angle turned by the wheel in time interval of t = 7.40 min
[tex]\theta = \omega t[/tex]
[tex]\theta = (\frac{2\pi}{37.3})(7.40) = 0.4\pi[/tex]
PART 1)
Now the average velocity is defined as the ratio of displacement and time
here displacement in given time interval is
[tex]d = 2Rsin\frac{\theta}{2}[/tex]
R = radius = 91.5 m
[tex]d = 183sin(0.2\pi) = 106.8 m[/tex]
Now time to turn the wheel is given as
[tex]t = 7.40 min = 444 s[/tex]
now we have
[tex]v = \frac{d}{t} = \frac{106.8}{444} [/tex]
[tex]v = 0.24 m/s[/tex]
PART 2)
Now average acceleration is defined as ratio of change in velocity in given time interval
here velocity of a point on its rim is given as
[tex]v = R\omega[/tex]
[tex]v = (91.5)(\frac{2\pi}{37.3\times 60})[/tex]
[tex]v = 0.257 m/s[/tex]
now change in velocity when wheel turned by the above mentioned angle is given as
[tex]\Delta v = 2vsin\frac{\theta}{2}[/tex]
[tex]\Delta v = 2(0.257)sin(0.2\pi)[/tex]
[tex]\Delta v = 0.3 m/s[/tex]
time interval is given as
[tex]t = 7.40 min = 444 s[/tex]
now average acceleration is given as
[tex]a = \frac{0.3}{444}[/tex]
[tex]a = 6.75 \times 10^{-4} m/s^2[/tex]
