Three astronauts, propelled by jet backpacks, push and guide a 149 kg asteroid toward a processing dock, exerting the forces shown in the figure, with F1 = 28 N, F2 = 59 N, F3 = 41 N, θ1 = 30°, and θ3 = 60°. What is the (a) magnitude and (b) angle (measured relative to the positive direction of the x axis in the range of (-180°, 180°]) of the asteroid's acceleration?
Well, it seems like those astronauts are really putting their back into it to move that asteroid! Let's calculate the magnitude and angle of the asteroid's acceleration.
To start, we'll need to resolve the forces into their x and y components.
Considering F1, the x-component, F1x, can be calculated as:
F1x = F1 * cos(θ1)
= 28 N * cos(30°)
= 24.2 N
And the y-component, F1y, can be calculated as:
F1y = F1 * sin(θ1)
= 28 N * sin(30°)
= 14 N
Similarly, let's calculate the x and y components for F2 and F3:
F2x = F2 * cos(180°)
= -59 N * cos(180°) (since cos(180°) = -1)
= 59 N
F2y = F2 * sin(180°)
= -59 N * sin(180°) (sin(180°) = 0)
= 0 N
F3x = F3 * cos(θ3)
= 41 N * cos(60°)
= 20.5 N
F3y = F3 * sin(θ3)
= 41 N * sin(60°)
= 35.5 N
Now that we have the x and y components of the forces, we can add them up to find the net force in the x and y direction:
Fnet_x = F1x + F2x + F3x
= 24.2 N + 59 N + 20.5 N
= 103.7 N
Fnet_y = F1y + F2y + F3y
= 14 N + 0 N + 35.5 N
= 49.5 N
The magnitude of the asteroid's acceleration can be calculated using the following formula:
a = sqrt(Fnet_x² + Fnet_y²)
= sqrt((103.7 N)² + (49.5 N)²)
= sqrt(10753.69 N² + 2450.25 N²)
= sqrt(13203.94 N²)
= 115.03 N
Great! We have the magnitude of the asteroid's acceleration, which is approximately 115.03 N.
To find the angle (θ) of the asteroid's acceleration, we can use the following formula:
θ = arctan(Fnet_y / Fnet_x)
= arctan(49.5 N / 103.7 N)
= arctan(0.477)
≈ 25.55°
So, the angle of the asteroid's acceleration, measured relative to the positive direction of the x-axis, is approximately 25.55°.
I hope that helped! Just remember, while those astronauts are working hard, it's important to keep an eye out for any space clowns trying to make things even more interesting!
To find the magnitude and angle of the asteroid's acceleration, we can use Newton's second law of motion, which states that the sum of all forces acting on an object is equal to the mass of the object multiplied by its acceleration.
Let's break down the forces acting on the asteroid into their x and y-components. Considering the x-direction as positive towards the right and the y-direction as positive upwards:
The x-component of force F1 is given by:
Fx1 = F1 * cos(θ1)
The x-component of force F3 is given by:
Fx3 = F3 * cos(θ3)
The total x-component of force is the sum of Fx1 and Fx3:
Fxtotal = Fx1 + Fx3
The y-component of force F1 is given by:
Fy1 = F1 * sin(θ1)
The y-component of force F3 is given by:
Fy3 = F3 * sin(θ3)
The total y-component of force is the sum of Fy1 and Fy3:
Fytotal = Fy1 + Fy3
Since the asteroid is being pushed towards the docking station, the net force will be in the negative x-direction. Therefore, Fxtotal will be negative.
Now, let's calculate the x and y-components of the total force and use them to find the acceleration.
Fx1 = F1 * cos(θ1) = 28 N * cos(30°) = 28 N * 0.866 = 24.248 N
Fx3 = F3 * cos(θ3) = 41 N * cos(60°) = 41 N * 0.5 = 20.5 N
Fxtotal = Fx1 + Fx3 = 24.248 N + 20.5 N = 44.748 N (this will be negative)
Fy1 = F1 * sin(θ1) = 28 N * sin(30°) = 28 N * 0.5 = 14 N
Fy3 = F3 * sin(θ3) = 41 N * sin(60°) = 41 N * 0.866 = 35.506 N
Fytotal = Fy1 + Fy3 = 14 N + 35.506 N = 49.506 N
The x-component of acceleration can be calculated using Newton's second law:
Fx = m * ax
∴ ax = Fxtotal / m
where m is the mass of the asteroid.
Given that the mass of the asteroid is 149 kg, we can substitute the values:
ax = (44.748 N) / (149 kg)
ax = 0.3 m/s²
The y-component of acceleration can be calculated in a similar manner:
Fy = m * ay
∴ ay = Fytotal / m
ay = (49.506 N) / (149 kg)
ay = 0.332 m/s²
Now that we have the x and y-components of the acceleration, we can find the magnitude and angle.
The magnitude of acceleration is given by the Pythagorean theorem:
|a| = sqrt(ax² + ay²)
|a| = sqrt((0.3 m/s²)² + (0.332 m/s²)²)
|a| = sqrt(0.09 m²/s⁴ + 0.109824 m²/s⁴)
|a| = sqrt(0.199824 m²/s⁴)
|a| ≈ 0.447 m/s²
The angle of acceleration can be found using the inverse tangent:
θ = atan(ay / ax)
θ = atan(0.332 m/s² / 0.3 m/s²)
θ = atan(1.1067)
θ ≈ 48.2°
Since the net force is in the negative x-direction, the angle will be negative. Therefore,
θ ≈ -48.2°
Therefore, the (a) magnitude of the asteroid's acceleration is approximately 0.447 m/s² and the (b) angle relative to the positive direction of the x-axis is approximately -48.2°.