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Two blocks, A and B, of masses 2m and 3m respectively, are attached to the ends of a light string - Edexcel - A-Level Maths Mechanics - Question 3 - 2019 - Paper 1
Question 3
Two blocks, A and B, of masses 2m and 3m respectively, are attached to the ends of a light string. Initially A is held at rest on a fixed rough plane. The plane is... show full transcript
Worked Solution & Example Answer:Two blocks, A and B, of masses 2m and 3m respectively, are attached to the ends of a light string - Edexcel - A-Level Maths Mechanics - Question 3 - 2019 - Paper 1
Step 1
Show that T = 12mg/5
Answer
To establish the tension T in the string, we apply Newton's second law for both blocks A and B.
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For block A (mass = 2m):
- The forces acting on A are the tension T, friction force F, and the component of the weight down the slope.
- The equation of motion can be expressed as:
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For block B (mass = 3m):
- The forces acting on B include its weight and the tension:
- The equation of motion is:
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Substituting F:
- Given that the coefficient of friction ( \frac{2}{3} ), friction F is:
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Relating angles:
- From the right triangle formed, we know ( \tan(\alpha) = \frac{5}{12} ).
- We can compute ( \sin(\alpha) ) and ( \cos(\alpha) ):
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Final steps:
- Expressing R:
- The tension equation thus combines into:
Step 2
Determine whether A will remain at rest, carefully justifying your answer.
Answer
After B reaches the ground, we must analyze the forces acting on A:
- Forces on A: The only forces acting on block A are the tension T in the string and the friction force F.
- Tension and Friction: When B reaches the ground, the tension will be less than it was during the motion. Thus, if the force due to friction exceeds the combined forces acting downward on A, A may remain at rest.
- Calculating Forces:
- The upward force due to friction must be calculated:
- The critical comparison is made between T and F. Since T reduces after B hits the ground, there’s a point where the forces may equalize allowing A to stay secure against slipping.
- Conclusion: If the resultant friction force is sufficient, A will remain at rest. If not, it will slide down the plane.
Step 3
Suggest two refinements to the model that would make it more realistic.
Answer
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Inclusion of Air Resistance: Adding air resistance to the model could provide a more accurate representation of the forces acting on both blocks, especially at higher velocities.
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Weight of the String: Considering the weight of the string itself could affect the tension in the string during the motion and provide a more precise calculation of dynamics as it would contribute additional downward force.