Answer:
The angle of rotation given is incorrect, the correct function to have a balanced unit is
θ(t) = (1.10 rad/s)t + (6.30 rad/s²)t²
Explanation:
Given that,
Radius of disk is
R = 0.4m
Mass of disk
M = 30kg
The angle of rotation is a function of time and it is given as
θ(t) = (1.10 rad/s)t + (6.30 rad/s²)t²
What is linear angular acceleration "a" when it rotates 0.1rev
θ = 0.1 rev
1 rev = 2πrad
θ = 0.1 × 2πrad = 0.63 rad
Let calculate the time it revolves 0.63rad
θ(t) = (1.10 rad/s)t + (6.30 rad/s²)t².
0.63 = 1.1t + 6.3t²
6.3t² + 1.1t - 0.63 = 0
Solving this quadratic equation using formula method
t = [-b±√(b²-4ac)]/2a
a = 6.3 b = 1.1 c = -0.63
t = [-1.1 ± √(1.1²-4×6.3×-0.63)]/2×6.3
t = -1.1±√(1.21+15.876) / 12.6
t = -1.1 ± 4.13 / 12.6
Let discard the negative time
So, t = (-1.1 + 4.13) / 12.6
t = 0.24 s
Now,
Angular velocity can be determine using
ωz = dθ / dt
ωz = 1.1 + 12.6t, since t = 0.24
ωz = 1.1 + 12.6(0.24)
ωz = 4.124 rad/s
Angular acceleration can be determine using
αz = dωz/dt
αz = 12.6 rad/s²
The radial acceleration can be determined using
ar = ωz²•R
ar = 4.124² × 0.4
ar = 6.8 m/s²
Also, the tangential Acceleration is related to angular acceleration using
at = αz • R
at = 12.6 × 0.4
at = 5.04 m/s²
Then, the magnitude of the resultant acceleration can be calculated using
a = √(ar² + at²)
a = √(6.8² + 5.04²)
a = √71.64
a = 8.46 m/s²
Direction
β = Arctan(at/ar)
β = Arctan(5.04/6.8)
β = 36.54
This is the direction from the origin
But it has already revolve 0.1rev
1 rev = 360°
Then, θ = 0.1° = 360 × 0.1 = 36°
Direction = β — θ
Then, it direction is 36.54 — 54° = 0.54°
The direction is 0.54°
Check attachment for direction understanding
1)
The capacitance of a parallel-plate capacitor is given by:
where
is the vacuum permittivity
A is the area of each plate
d is the distance between the plates
Here, the radius of each plate is
so the area is
While the separation between the plates is
So the capacitance is
And now we can find the energy stored,which is given by:
2) 0.71 J/m^3
The magnitude of the electric field is given by
and the energy density of the electric field is given by
and using
, we find
Answer:
41°
Explanation:
Kinetic energy at bottom = potential energy at top
½ mv² = mgh
½ v² = gh
h = v²/(2g)
h = (2.4 m/s)² / (2 × 9.8 m/s²)
h = 0.294 m
The pendulum rises to a height of above the bottom. To determine the angle, we need to use trigonometry (see attached diagram).
L − h = L cos θ
cos θ = (L − h) / L
cos θ = (1.2 − 0.294) / 1.2
θ = 41.0°
Rounded to two significant figures, the pendulum makes a maximum angle of 41° with the vertical.
