Two forces are pushing on an object, one at 12 lbs of Force and one at 5.66 lbs of Force. The angle between them is 35° (each is 72.5 from horizontal, such that the forces make a v with the object in the center).
17. What is the total Force on the object?
18. What is the smallest angle of the triangle?
19. What is the largest angle in the triangle?
20. What is the remaining angle of the triangle?
To find the total force on the object, we need to calculate the vector sum of the two forces. We can use the law of cosines to find the magnitude of the total force. The law of cosines states:
c² = a² + b² - 2ab*cos(C)
where c is the side opposite to angle C, and a and b are the sides adjacent to angle C. In this case, the magnitudes of the two forces are a = 12 lbs and b = 5.66 lbs, and the angle between them is C = 35°.
Using the formula, we can calculate the magnitude of the total force:
c² = (12²) + (5.66²) - 2(12)(5.66)*cos(35°)
c² = 144 + 32 - 2(12)(5.66)*cos(35°)
c² = 176 - 135.84*cos(35°)
c² ≈ 176 - 99.656 ≈ 76.344
Taking the square root of both sides, we find:
c ≈ √76.344 ≈ 8.74
Therefore, the total force on the object is approximately 8.74 lbs.
To find the smallest angle of the triangle, we can use the law of sines. The law of sines states:
a/sin(A) = b/sin(B) = c/sin(C)
where a, b, and c are the magnitudes of the sides opposite to angles A, B, and C, respectively.
In this case, we know the magnitudes of two sides, a = 12 lbs and b = 5.66 lbs, and the opposite angles are A = 72.5° and B = 72.5°. We can solve for the smallest angle C using the law of sines.
a/sin(A) = b/sin(B) = c/sin(C)
12/sin(72.5°) = 5.66/sin(72.5°) = 8.74/sin(C)
sin(C) = (8.74/sin(A))*sin(C) = (8.74/sin(72.5°))*sin(72.5°) = 8.74
C ≈ arcsin(8.74)
Therefore, the smallest angle of the triangle is approximately arcsin(8.74).
To find the largest angle in the triangle, we can subtract the sum of the two known angles (72.5° and 72.5°) from 180°.
180° - (72.5° + 72.5°) = 180° - 145° = 35°
Therefore, the largest angle in the triangle is 35°.
To find the remaining angle of the triangle, we can subtract the sum of the three known angles (72.5°, 72.5°, and 35°) from 180°.
180° - (72.5° + 72.5° + 35°) = 180° - 180° = 0°
Therefore, the remaining angle of the triangle is 0°.
To solve these questions, we'll be using the concept of vector addition and the properties of a triangle. Here's how we can find the answers:
17. To find the total force on the object, we need to add the two forces together. Since the forces are not in the same direction, we'll use vector addition to find the resultant force. We can break each force into its horizontal and vertical components.
First, let's find the horizontal component of each force:
Horizontal component of 12 lbs force = 12 lbs * cos(72.5°)
Horizontal component of 5.66 lbs force = 5.66 lbs * cos(72.5°)
Next, let's find the vertical component of each force:
Vertical component of 12 lbs force = 12 lbs * sin(72.5°)
Vertical component of 5.66 lbs force = 5.66 lbs * sin(72.5°)
Now, add the horizontal and vertical components separately to get the total horizontal and vertical components of the resultant force.
Total horizontal component = Sum of horizontal components of the forces
Total vertical component = Sum of vertical components of the forces
Finally, use the Pythagorean theorem to find the magnitude of the resultant force:
Total Force = sqrt((Total horizontal component)^2 + (Total vertical component)^2)
18. To find the smallest angle of the triangle, we can look at the angles formed between the object and each of the forces. We can use the property that the smallest angle in a triangle is opposite to the smallest side. In this case, the smallest angle would be opposite to the 5.66 lbs force.
19. To find the largest angle in the triangle, we can use the property that the largest angle is opposite to the largest side. In this case, the largest angle would be opposite to the total force.
20. Since a triangle's angles add up to 180°, we can find the remaining angle by subtracting the sum of the other two angles (smallest angle and largest angle) from 180°.