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A small stone A of mass 3m is attached to one end of a string - Edexcel - A-Level Maths: Mechanics - Question 2 - 2021 - Paper 1
Question 2
A small stone A of mass 3m is attached to one end of a string. A small stone B of mass m is attached to the other end of the string. Initially A is held at rest on a... show full transcript
Worked Solution & Example Answer:A small stone A of mass 3m is attached to one end of a string - Edexcel - A-Level Maths: Mechanics - Question 2 - 2021 - Paper 1
Step 1
write down an equation of motion for A
Answer
To derive the equation of motion for stone A, we can consider the forces acting on it. The forces include:
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The gravitational force along the plane:
F_g = 3mg imes rac{3}{5}
where the 3 in the numerator represents mass A (3m) and the angle of incline is given by the tangent function.
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The frictional force opposing the motion:
F_f = rac{1}{6} R
where R is the normal reaction force, defined as:
R = 3mg imes rac{4}{5}
using the cosine of angle α to resolve it perpendicular to the plane.
The equation of motion can be expressed as:
3mg imes rac{3}{5} - F_f - T = 3ma
Thus, the equation becomes:
3mg imes rac{3}{5} - rac{1}{6} R - T = 3ma.
Step 2
show that the acceleration of A is \frac{1}{10}g
Answer
First, we know from the previous part the equation:
3mg imes rac{3}{5} - \frac{1}{6} R - T = 3ma
Next, substituting the value of R:
When substituted, we derive:
Resolving the equation further yields:
Thus on simplifying:
Therefore, the acceleration a can be expressed in terms of g as:
$$a = \frac{1}{10}g.$
Step 3
sketch a velocity-time graph for the motion of B, from the instant when A is released from rest to the instant just before B reaches the pulley, explaining your answer
Answer
The motion of stone B can be visualized on a velocity-time graph of the form:
- Initial conditions: When stone A is released, stone B begins with zero velocity (u = 0).
- Acceleration: Since the acceleration of A is constant at \frac{1}{10}g, this also affects the velocity of B as the two stones are connected.
- Graph representation: The graph starts at the origin (0,0) and rises linearly with a positive slope, representing constant acceleration until B reaches the pulley.
- Final velocity: The final velocity right before B reaches the pulley is the last point on the graph, marked based on the time taken to reach this distance.
In conclusion, the velocity-time graph will be a straight line for the motion of B, depicting continuous acceleration.
Step 4
State how this would affect the working of part (b)
Answer
The working of part (b) would be affected as follows:
- Since stone A is accelerating due to gravity, it affects tension experienced by both stones.
- The equation derived is dependent on the uniform acceleration of stone A.
- If the system is not ideal (e.g., friction was higher or the string was not inextensible), the derived acceleration value would change.
- Therefore, any change in the pulling force or resistance would alter the effectiveness of the calculated acceleration of stone A, leading to inaccuracies in applying the formula derived for the motion.