To find the sum of the forces on the piano when Jacob pushes with 50 N to the east and Sam pushes with 70 N to the north, we need to combine these forces as vectors. Here are the three required answers:

1. Magnitude of the Resultant Force:

To find the magnitude $F_{\text{resultant}}$ of the combined forces, we can use the Pythagorean theorem, since the forces are perpendicular to each other (east and north directions).

$F_{\text{resultant}} = \sqrt{(F_{\text{Jacob}})^2 + (F_{\text{Sam}})^2}$

Substitute the given values:

$F_{\text{resultant}} = \sqrt{(50)^2 + (70)^2}$

$F_{\text{resultant}} = \sqrt{2500 + 4900} = \sqrt{7400} \approx 86.02 \, \text{N}$

So, the magnitude of the resultant force is approximately 86.02 N.

---

2. Direction of the Resultant Force (Angle θ):

To find the direction (angle θ) of the resultant force relative to the east (Jacob's force direction), we use the tangent function:

$\tan(\theta) = \frac{F_{\text{Sam}}}{F_{\text{Jacob}}}$

Substitute the given values:

$\tan(\theta) = \frac{70}{50} = 1.4$

Now, calculate the angle:

$\theta = \tan^{-1}(1.4) \approx 54.46^\circ$

So, the direction of the resultant force is approximately 54.46° north of east.

---

3. The Resultant Force as a Vector:

We can express the resultant force as a vector in component form, where the force has an eastward and northward component.

$\vec{F}_{\text{resultant}} = (50 \, \text{N}, 70 \, \text{N})$

Thus, the resultant force vector is (50 N, 70 N), meaning 50 N in the east direction and 70 N in the north direction.

---

Let me know if you'd like further details or have any questions!

5 Related Questions:

1. How would the resultant force change if Jacob pushed with 100 N instead of 50 N?
2. What happens to the angle if Sam increases his force to 100 N while Jacob's force remains 50 N?
3. How do you resolve forces when they are not perpendicular to each other?
4. What is the significance of breaking forces into components in physics?
5. How would you calculate the work done if the piano moves 10 meters in the direction of the resultant force?

Tip:

When dealing with forces in different directions, always break them down into perpendicular components (like east and north) to make calculations easier!