Question

4. Determine the angle between the force F = (2i+ 5j) N and the displacement 5 = (-i+ 7) m. The work done in applying the force over the displacement is 33 J. A. 17.4° B. 21.6° C. 29.9° D. 32.7° E. 39.8°​

Answer 1

Step-by-step explanation:

The work done by a force F over a displacement d is given by the dot product of the force and the displacement:

W = F · d

where · denotes the dot product.

We are given that the work done is 33 J, so

F · d = 33 J

We can find the dot product by taking the sum of the products of the corresponding components:

F · d = (2i + 5j) · (-i + 7j) = 2(-1) + 5(7) = 33

Now, we can find the magnitude of the force and displacement vectors:

|F| = √(2² + 5²) = √29

|d| = √((-1)² + 7²) = √50

The angle between the force and displacement vectors can be found using the dot product:

F · d = |F| |d| cos θ

cos θ = (F · d) / (|F| |d|)

cos θ = 33 / (√29 √50)

cos θ ≈ 0.603

θ ≈ 53.2°

However, we are looking for the angle between the force and the displacement, which is the supplement of θ, so:

angle = 180° - 53.2° ≈ 126.8°

None of the answer choices match this result, so we must have made a mistake.

Let's check our work. We made an error in the calculation of the cosine of θ. It should be:

cos θ = (F · d) / (|F| |d|)

cos θ = 33 / (√29 √50)

cos θ ≈ 0.433

θ ≈ 64.2°

The angle between the force and displacement vectors is approximately 64.2°.

Now we can check the answer choices:

A. 17.4° - Too small

B. 21.6° - Too small

C. 29.9° - Too small

D. 32.7° - Too small

E. 39.8° - Too small

None of the answer choices match the calculated angle of 64.2°. Therefore, the correct answer is not among the choices given.

Answer 2

Answer:

option C

Step by step Explanation:

The work done by a constant force F acting on an object over a displacement d is given by the dot product of F and d:

W = F · d = |F| |d| cosθ

where |F| and |d| are the magnitudes of F and d, respectively, and θ is the angle between F and d.

We are given that F = (2i+ 5j) N, d = (-i+ 7j) m, and W = 33 J. We can first find the magnitudes of F and d:

|F| = √(2^2 + 5^2) N ≈ 5.39 N

|d| = √((-1)^2 + 7^2) m ≈ 7.07 m

We can then solve for cosθ:

cosθ = W / (|F| |d|) = 33 J / (5.39 N × 7.07 m) ≈ 0.849

Finally, we can find the angle θ:

θ = cos⁻¹(0.849) ≈ 29.9°

Therefore, the angle between the force F and the displacement d is approximately 29.9°, which is option C