Juana and Diego Gonzales, ages six and four respectively, own a strong and stubborn puppy named Corporal. It is so hard to take Corporal for a walk that they devise a scheme to use two leashes. If Juana and Diego pull with forces of 23 lb and 27 lb at the angles shown in the figure (Juana at 18 degrees from the horizontal, Diego at 15 degrees the other side from the horizontal), how hard is Corporal pulling if the puppy holds the children at a standstill?
23 cos 18 + 27 cos 15
Thank you!
To find out how hard Corporal is pulling, we need to combine the forces exerted by Juana and Diego using vector addition. First, let's break down each force into its horizontal and vertical components.
The horizontal component of Juana's force (F1) can be calculated as:
F1_horizontal = F1 * cosθ1
where F1 is the magnitude of Juana's force (23 lb) and θ1 is the angle it makes with the horizontal (18 degrees).
Substituting the values, we get:
F1_horizontal = 23 lb * cos(18°)
F1_horizontal ≈ 22.223 lb
Similarly, the horizontal component of Diego's force (F2) can be calculated as:
F2_horizontal = F2 * cosθ2
where F2 is the magnitude of Diego's force (27 lb) and θ2 is the angle it makes with the horizontal (15 degrees).
Substituting the values, we get:
F2_horizontal = 27 lb * cos(15°)
F2_horizontal ≈ 26.362 lb
Now, let's find the total horizontal component of the force exerted by both Juana and Diego (F_total_horizontal):
F_total_horizontal = F1_horizontal + F2_horizontal
F_total_horizontal ≈ 22.223 lb + 26.362 lb
F_total_horizontal ≈ 48.585 lb
Next, let's calculate the vertical component of Juana's force (F1_vertical):
F1_vertical = F1 * sinθ1
Substituting the values, we get:
F1_vertical = 23 lb * sin(18°)
F1_vertical ≈ 6.526 lb
Similarly, the vertical component of Diego's force (F2_vertical) can be calculated as:
F2_vertical = F2 * sinθ2
Substituting the values, we get:
F2_vertical = 27 lb * sin(15°)
F2_vertical ≈ 7.027 lb
Now, let's find the total vertical component of the force exerted by both Juana and Diego (F_total_vertical):
F_total_vertical = F1_vertical + F2_vertical
F_total_vertical ≈ 6.526 lb + 7.027 lb
F_total_vertical ≈ 13.553 lb
Lastly, let's calculate the magnitude of the force exerted by Corporal using the Pythagorean theorem:
F_total = √( F_total_horizontal^2 + F_total_vertical^2 )
Substituting the values, we get:
F_total = √( 48.585 lb^2 + 13.553 lb^2 )
F_total ≈ √( 2358.657 lb^2 )
F_total ≈ 48.562 lb
Therefore, Corporal is pulling with a force of approximately 48.562 lb.
To determine how hard Corporal is pulling, we need to find the resultant force acting on the puppy by adding the horizontal and vertical components of the forces applied by Juana and Diego.
Let's break down the forces applied by Juana and Diego into their horizontal and vertical components:
The horizontal component of Juana's force:
F_hj = F_j * cos(18 degrees)
where F_j is Juana's force of 23 lb.
Substituting the values, we have:
F_hj = 23 lb * cos(18 degrees)
The vertical component of Juana's force:
F_vj = F_j * sin(18 degrees)
where F_j is Juana's force of 23 lb.
Substituting the values, we have:
F_vj = 23 lb * sin(18 degrees)
Note: The negative sign for F_vj is because the force acts downwards.
The horizontal component of Diego's force:
F_hd = F_d * cos(15 degrees)
where F_d is Diego's force of 27 lb.
Substituting the values, we have:
F_hd = 27 lb * cos(15 degrees)
Note: The negative sign for F_hd is because the force acts in the opposite direction.
The vertical component of Diego's force:
F_vd = F_d * sin(15 degrees)
where F_d is Diego's force of 27 lb.
Substituting the values, we have:
F_vd = 27 lb * sin(15 degrees)
Now, let's add the horizontal and vertical forces to find the resultant force:
The horizontal component of the resultant force:
F_hr = F_hj + F_hd
Substituting the values, we have:
F_hr = 23 lb * cos(18 degrees) + 27 lb * cos(15 degrees)
The vertical component of the resultant force:
F_vr = F_vj + F_vd
Substituting the values, we have:
F_vr = 23 lb * sin(18 degrees) + 27 lb * sin(15 degrees)
Now, we can calculate the magnitude and direction of the resultant force.
The magnitude of the resultant force, F_r, can be found using the Pythagorean theorem:
F_r = sqrt(F_hr^2 + F_vr^2)
Substituting the values, we have:
F_r = sqrt((23 lb * cos(18 degrees) + 27 lb * cos(15 degrees))^2 + (23 lb * sin(18 degrees) + 27 lb * sin(15 degrees))^2)
The direction of the resultant force can be found using the inverse tangent:
θ = atan(F_vr / F_hr)
Substituting the values, we have:
θ = atan((23 lb * sin(18 degrees) + 27 lb * sin(15 degrees)) / (23 lb * cos(18 degrees) + 27 lb * cos(15 degrees)))
Therefore, by calculating the values of F_r and θ, we can determine how hard Corporal is pulling to keep Juana and Diego at a standstill.
Note: The above calculations assume that the forces are acting in a two-dimensional plane and that the angles given are with respect to the horizontal axis.