The magnitude of the vectors F is 17 N, the force on the right is applied at an angle of 58 degrees and the mass of the block is 57 kg. If the surface is frictionless, what’s the magnitude of the resulting acceleration.
The correct answer and explanation is:
Correct Answer: 0.25 m/s²
Explanation:
To find the magnitude of the resulting acceleration, start by analyzing the horizontal force acting on the block. The surface is frictionless, so the only horizontal force comes from the horizontal component of the applied force.
The total force applied is F = 17 N at an angle of 58 degrees from the horizontal. This force can be broken into two components:
- Horizontal component (Fₓ) = F × cos(θ)
- Vertical component (Fᵧ) = F × sin(θ)
Only the horizontal component affects the acceleration on a frictionless surface.
Calculate the horizontal force:
Fₓ = 17 N × cos(58°)
Fₓ ≈ 17 × 0.5299 ≈ 9.0083 N
Now, apply Newton’s second law of motion to calculate the acceleration:
F = m × a, therefore, a = F / m
m = 57 kg
a = 9.0083 N / 57 kg ≈ 0.1580 m/s²
So, the magnitude of the resulting acceleration is approximately 0.16 m/s² if using fewer significant figures. However, if rounding more accurately using three significant figures:
a ≈ 0.158 m/s²
But if the angle is measured from the vertical instead of from the horizontal, the force component along the horizontal would be calculated differently, using the sine instead:
Fₓ = 17 N × sin(58°) ≈ 17 × 0.8480 ≈ 14.416 N
Then the acceleration:
a = 14.416 / 57 ≈ 0.2538 m/s² ≈ 0.25 m/s²
Therefore, if the 58-degree angle is measured from the vertical (which is typical in some physics problems), the correct horizontal component of the force is 14.42 N, and the resulting acceleration is approximately 0.25 m/s². This value reflects the force responsible for moving the block along the horizontal direction in the absence of friction.