Answer:
8.91 J
Explanation:
mass, m = 8.20 kg
radius, r = 0.22 m
Moment of inertia of the shell, I = 2/3 mr^2
= 2/3 x 8.2 x 0.22 x 0.22 = 0.265 kgm^2
n = 6 revolutions
Angular displacement, θ = 6 x 2 x π = 37.68 rad
angular acceleration, α = 0.890 rad/s^2
initial angular velocity, ωo = 0 rad/s
Let the final angular velocity is ω.
Use third equation of motion
ω² = ωo² + 2αθ
ω² = 0 + 2 x 0.890 x 37.68
ω = 8.2 rad/s
Kinetic energy,
K = 0.5 x 0.265 x 8.2 x 8.2
K = 8.91 J
Answer:
a) L=0. b) L = 262 k ^ Kg m²/s and c) L = 1020.7 k^ kg m²/s
Explanation:
It is angular momentum given by
L = r x p
Bold are vectors; where L is the angular momentum, r the position of the particle and p its linear momentum
One of the easiest ways to make this vector product is with the use of determinants
Let's apply this relationship to our case
Let's start by breaking down the speed
v₀ₓ = v₀ cosn 45
voy =v₀ sin 45
v₀ₓ = 9 cos 45
voy = 9 without 45
v₀ₓ = 6.36 m / s
voy = 6.36 m / s
a) at launch point r = 0 whereby L = 0
. b) let's find the position for maximum height, we can use kinematics, at this point the vertical speed is zero
vfy² = voy²- 2 g y
y = voy² / 2g
y = (6.36)²/2 9.8
y = 2.06 m
Let's calculate the angular momentum
L=
L = -px y k ^
L = - (m vox) (2.06) k ^
L = - 20 6.36 2.06 k ^
L = 262 k ^ Kg m² / s
The angular momentum is on the z axis
c) At the point of impact, at this point the height is zero and the position on the x-axis is the range
R = vo² sin 2θ / g
R = 9² sin (2 45) /9.8
R = 8.26 m
L =
L = - x py k ^
L = - x m voy
L = - 8.26 20 6.36 k ^
L = 1020.7 k^ kg m² /s
Answer:
His launching angle was 14.72°
Explanation:
Please, see the figure for a graphic representation of the problem.
In a parabolic movement, the velocity and displacement vectors are two-component vectors because the object moves along the horizontal and vertical axis.
The horizontal component of the velocity is constant, while the vertical component has a negative acceleration due to gravity. Then, the velocity can be written as follows:
v = (vx, vy)
where vx is the component of v in the horizontal and vy is the component of v in the vertical.
In terms of the launch angle, each component of the initial velocity can be written using the trigonometric rules of a right triangle (see attached figure):
sin angle = opposite / hypotenuse
cos angle = adjacent / hypotenuse
In our case, the side opposite the angle is the module of v0y and the side adjacent to the angle is the module of vx. The hypotenuse is the module of the initial velocity (v0). Then:
sin angle = v0y / v0 then: v0y = v0 * sin angle
In the same way for vx:
vx = v0 * cos angle
Using the equation for velocity in the x-axis we can find the equation for the horizontal position:
dx / dt = v0 * cos angle
dx = (v0 * cos angle) dt (integrating from initial position, x0, to position at time t and from t = 0 and t = t)
x - x0 = v0 t cos angle
x = x0 + v0 t cos angle
For the displacement in the y-axis, the velocity is not constant because the acceleration of the gravity:
dvy / dt = g ( separating variables and integrating from v0y and vy and from t = 0 and t)
vy -v0y = g t
vy = v0y + g t
vy = v0 * sin angle + g t
The position will be:
dy/dt = v0 * sin angle + g t
dy = v0 sin angle dt + g t dt (integrating from y = y0 and y and from t = 0 and t)
y = y0 + v0 t sin angle + 1/2 g t²
The displacement vector at a time "t" will be:
r = (x0 + v0 t cos angle, y0 + v0 t sin angle + 1/2 g t²)
If the launching and landing positions are at the same height, then the displacement vector, when the object lands, will be (see figure)
r = (x0 + v0 t cos angle, 0)
The module of this vector will be the the total displacement (65 m)
module of r =
65 m = x0 + v0 t cos angle ( x0 = 0)
65 m / v0 cos angle = t
Then, using the equation for the position in the y-axis:
y = y0 + v0 t sin angle + 1/2 g t²
0 = y0 + v0 t sin angle + 1/2 g t²
replacing t = 65 m / v0 cos angle and y0 = 0
0 = 65m (v0 sin angle / v0 cos angle) + 1/2 g (65m / v0 cos angle)²
cancelating v0:
0 = 65m (sin angle / cos angle) + 1/2 g * (65m)² / (v0² cos² angle)
-65m (sin angle / cos angle) = 1/2 g * (65m)² / (v0² cos² angle)
using g = -9.8 m/s²
-(sin angle / cos angle) * (cos² angle) = -318.5 m²/ s² / v0²
sin angle * cos angle = 318.5 m²/ s² / (36 m/s)²
(using trigonometric identity: sin x cos x = sin (2x) / 2
sin (2* angle) /2 = 0.25
sin (2* angle) = 0.49
2 * angle = 29.44
<u>angle = 14.72°</u>
