A 19 kg mass, slides down a rough inclined plane at an angle of 32.5 to the horizontal, experiencing a frictional force of 6.7 N.Determine the acceleration of the mass.

Question

BlackTom AI · Accepted Answer

The component of gravity pulling the block down the plane is mg sinθ, so start with that.

1) Relevant concepts
- Decompose weight into components: parallel to plane = mg sinθ, perpendicular = mg cosθ.
- Net force down the plane = mg sinθ − friction.
- Acceleration = (net force)/m (Newton’s second law).

2) Step-by-step reasoning and calculations
- m = 19.0 kg, θ = 32.5°, friction force f = 6.7 N, g = 9.80 m/s^2.
- Compute weight magnitude: mg = 19.0 × 9.80 = 186.2 N.
- Compute sin(32.5°). sin 32.5° ≈ 0.5373.
- Gravitational component down the plane: mg sinθ = 186.2 × 0.5373 ≈ 100.06 N.
- Subtract friction to get net force down the plane: F_net = 100.06 − 6.7 = 93.36 N.
- Use Newton’s second law: a = F_net / m = 93.36 / 19.0 = 4.9137 m/s^2.
- Round to three significant figures: a ≈ 4.92 m/s^2.

3) Intermediate steps shown above (mg, sinθ, mg sinθ, subtraction of friction, division by m).

4) Why the correct answer is correct
- The calculation properly resolves gravity along the incline, subtracts the given frictional force acting opposite the motion, and divides by the mass to obtain acceleration. All units are consistent and the arithmetic yields 4.92 m/s^2.

5) Why the other options are incorrect
- (c) 5.27 m/s^2 equals g sinθ (≈9.80×sin32.5) and therefore corresponds to ignoring friction — it’s too large because it omits the 6.7 N opposing force.
- (b) 7.92 m/s^2 and (d) 8.27 m/s^2 are substantially larger than the correct result and do not match any correct decomposition or application of Newton’s second law for this problem (they reflect arithmetic or conceptual errors such as using the full weight or incorrect angle/component).

6) Conclusion
The calculated acceleration is 4.92 m/s^2, which matches the provided correct answer (option a).

A 19 kg mass, slides down a rough inclined plane at an angle of 32.5 to the horizontal, experiencing a frictional force of 6.7 N.Determine the acceleration of the mass.单项选择题

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