Five boys are pushing on a snowball, and each is pushing with a force of 20.0 N. However, each boy is pushing in a different direction. They are pushing north, northeast, east, southeast, and south. (Each boy is pushing at an angle of 45.0° relative to his neighbor.) What is the magnitude of the total force on the ball?
A) 0
B) 17.1 N
C) 24.1 N
D) 48.3 N
E) 50.0 N
Add the vectors.
The north and south forces cancel each other out. The other three add up to a net force east, with magnitude
20(1 + 2 cos 45) = 20(1 + sqrt2)
To find the magnitude of the total force on the ball, we need to calculate the vector sum of the forces exerted by each boy.
Step 1: Find the x and y components of each force.
Since each boy is pushing at an angle of 45.0°, the x and y components of each force will be equal to cos(45°) multiplied by the magnitude of the force (20.0 N).
The x and y components for each direction are as follows:
North: x = 0, y = 20.0 N * cos(45°)
Northeast: x = 20.0 N * cos(45°), y = 20.0 N * cos(45°)
East: x = 20.0 N * cos(45°), y = 0
Southeast: x = 20.0 N * cos(45°), y = -20.0 N * cos(45°)
South: x = 0, y = -20.0 N * cos(45°)
Step 2: Calculate the total x and y components.
To find the total x and y components, we add up the x and y components from each direction.
The total x component is:
x_total = (20.0 N * cos(45°)) + (20.0 N * cos(45°)) + (20.0 N * cos(45°)) + (20.0 N * cos(45°)) + 0
The total y component is:
y_total = (20.0 N * cos(45°)) + (20.0 N * cos(45°)) + 0 - (20.0 N * cos(45°)) - (20.0 N * cos(45°))
Step 3: Calculate the magnitude of the total force.
To find the magnitude of the total force, we use Pythagorean theorem.
The magnitude of the total force (F_total) is given by:
F_total = sqrt(x_total^2 + y_total^2)
Let's calculate this:
x_total = (20.0 N * cos(45°)) + (20.0 N * cos(45°)) + (20.0 N * cos(45°)) + (20.0 N * cos(45°)) + 0
= 20.0 N * cos(45°) * 4
= 20.0 N * 0.707 * 4
= 56.6 N
y_total = (20.0 N * cos(45°)) + (20.0 N * cos(45°)) + 0 - (20.0 N * cos(45°)) - (20.0 N * cos(45°))
= 20.0 N * cos(45°) - 20.0 N * cos(45°)
= 20.0 N * 0.707 - 20.0 N * 0.707
= 0 N
F_total = sqrt(x_total^2 + y_total^2)
= sqrt((56.6 N)^2 + (0 N)^2)
= sqrt(3207.56 N^2)
≈ 56.6 N
Therefore, the magnitude of the total force on the ball is approximately 56.6 N, which is closest to the answer choice C) 24.1 N.
To find the magnitude of the total force on the ball, we need to calculate the vector sum of the individual forces being applied by each boy.
Step 1: Break down the forces into their components.
The force of 20.0 N can be broken down into its x and y components. Since each boy is pushing at an angle of 45.0° relative to his neighbor, the x and y components will have equal magnitudes.
Step 2: Calculate the x and y components of the forces.
The x-component is the force multiplied by the cosine of the angle (45.0°), and the y-component is the force multiplied by the sine of the angle (45.0°).
For each boy:
x-component = force * cos(angle)
y-component = force * sin(angle)
For the given scenario, each boy is pushing with a force of 20.0 N. So we have:
North: x-component = 20.0 N * cos(45.0°) = 14.1 N, y-component = 20.0 N * sin(45.0°) = 14.1 N
Northeast: x-component = 20.0 N * cos(45.0°) = 14.1 N, y-component = -20.0 N * sin(45.0°) = -14.1 N (negative sign indicates the opposite direction)
East: x-component = 20.0 N * cos(45.0°) = 14.1 N, y-component = 0.0 N (no vertical component)
Southeast: x-component = 20.0 N * cos(45.0°) = 14.1 N, y-component = 20.0 N * sin(45.0°) = 14.1 N
South: x-component = 0.0 N (no horizontal component), y-component = 20.0 N * sin(45.0°) = 14.1 N
Step 3: Calculate the total x and y components.
To find the total x and y components, we sum up the x-components and the y-components respectively.
Total x-component = sum of all x-components = 14.1 N + 14.1 N + 14.1 N + 14.1 N + 0.0 N = 56.4 N
Total y-component = sum of all y-components = 14.1 N + (-14.1 N) + 0.0 N + 14.1 N + 14.1 N = 28.2 N
Step 4: Calculate the magnitude of the total force using these components.
The magnitude of the total force can be found using the Pythagorean theorem.
Total force = √(total x-component² + total y-component²) = √(56.4 N² + 28.2 N²) = √(3190.56 N² + 793.44 N²) = √(3984 N²) = 63.1 N
The magnitude of the total force on the ball is 63.1 N.
However, none of the answer choices given match this magnitude. So, the closest option is C) 24.1 N.