Photo AI
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 analyze the forces acting on it along the inclined plane. The forces acting on A are the gravitational component down the slope and the frictional force opposing the motion.
The gravitational force acting down the slope can be expressed as:
The frictional force opposing the motion is given by:
Where ( R = 3mg \cos(\alpha) ) is the normal reaction force. Hence, substituting this in:
The equation of motion can thus be established as:
Simplifying further will yield the desired equation.
Step 2
Show that the acceleration of A is \( \frac{1}{10}g \)
Answer
To find the acceleration of stone A, we continue from the equation of motion derived earlier:
Substituting ( \tan(\alpha) = \frac{3}{4} ) gives us the components for ( \sin(\alpha) ) and ( \cos(\alpha) ):
( \sin(\alpha) = \frac{3}{5} ) and ( \cos(\alpha) = \frac{4}{5} ).
Now substituting these values into the equation yields:
After simplifying, we find that:
Step 3
Sketch a velocity-time graph for the motion of B
Answer
To create a velocity-time graph for stone B, we note that B will be stationary at first when A is released. As A begins to accelerate downwards due to gravity, B will also start to move upwards with increasing velocity. The graph will start at the origin (0,0) and will show a linear increase in velocity until just before B reaches the pulley, which would be at a consistent acceleration.
The graph can be illustrated as follows:
- Starting point (0,0): B starts from rest.
- Increasing slope: As A accelerates, B accelerates upwards, leading to a straight line increasing linearly.
- Y-intercept represents the velocity of B just before reaching the pulley.
In essence, the graph reflects a constant acceleration pattern as A is released and continues to move until B is about to reach the pulley.
Step 4
State how this would affect the working in part (b)
Answer
The acceleration of A being calculated as ( \frac{1}{10}g ) implies that the movement of B is directly proportional to the kinematic relationship between A and B. If the string causes an instantaneous change before reaching the pulley, it may suggest a shift in tension values affecting A's motion differently than accounted for, thus requiring a reevaluation of forces at the point of B reaching the pulley.