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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
a) write down an equation of motion for A
Answer
To derive the equation of motion for A, we start by analyzing the forces acting on stone A. The forces acting on A are the gravitational force component along the plane, tension in the string, and friction. The equation can be written as:
Where:
- ( \sin(\alpha) = \frac{3}{5} ) (derived from ( tan(\alpha) = \frac{3}{4} ) using the Pythagorean identity)
- ( F = \frac{1}{6}R )
- R is the normal reaction force acting on A.
Step 2
b) show that the acceleration of A is \( \frac{1}{10} g \)
Answer
To show that the acceleration ( a ) of A is ( \frac{1}{10} g ), we proceed with the following calculations:
-
Resolve the forces:
- Normal force: ( R = 3mg \cos(\alpha) = 3mg \cdot \frac{4}{5} = \frac{12}{5}mg )
- Friction: ( F = \frac{1}{6}R = \frac{1}{6} \cdot \frac{12}{5}mg = \frac{2}{5}mg )
-
Substitute back into the equation of motion:
- Plugging in the normal and frictional forces:
- ( 3mg \cdot \frac{3}{5} - \frac{2}{5}mg - T = 3ma ) leads to:
- ( rac{9}{5}mg - \frac{2}{5}mg - T = 3ma )
- Simplifying this gives:
- ( T = \frac{7}{5}mg - 3ma )
-
For the whole system:
- The equations for B and A must yield:
- ( mg - T = ma )
- Combine and solve to find:
- ( a = \frac{1}{10} g ).
Step 3
c) sketch a velocity-time graph for the motion of B
Answer
The velocity-time graph for stone B would begin at rest (zero velocity) when A is released. Initially, as A descends, B begins to move upwards, gaining velocity. Since the acceleration is constant until B reaches the pulley, the graph will be linear, showing a steady increase in velocity until it reaches its maximum right before the pulley. The slope of this linear graph represents the acceleration of B, which is equal to the acceleration of A (( \frac{1}{10} g )).
Step 4
d) State how this would affect the working in part (b)
Answer
If the string is not light, the tension in the string would not be constant throughout the motion and would depend on the mass of the string itself. This would introduce an additional force component that must be considered in the equations of motion. Consequently, the effective acceleration calculated in part (b) would differ, leading to a different relationship between forces, ultimately affecting the derived acceleration of stone A.