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

To show that the acceleration of A is ( \frac{1}{10} g ), we start from the modified equation of motion derived in part (a). First, we need to derive expressions for ( \sin \alpha ) and ( \cos \alpha ).

Given ( \tan \alpha = \frac{3}{4} ), we can find:

$$\sin \alpha = \frac{3}{5}, \quad \cos \alpha = \frac{4}{5}.$$

Substituting these values into the equation:

$$3mg \frac{3}{5} - \frac{1}{6} \cdot 3mg \frac{4}{5} - T = 3ma$$

Now calculating each term:

• The force component: ( 3mg \frac{3}{5} = \frac{9mg}{5} )
• The frictional force: ( \frac{1}{6} \cdot 3mg \frac{4}{5} = \frac{2mg}{5} )

Thus, substituting:

$$\frac{9mg}{5} - \frac{2mg}{5} - T = 3ma$$

From here, simplifying gives:

\frac{7mg}{5} - T = 3ma $$. To isolate T, we rearrange:

T = \frac{7mg}{5} - 3ma

$$In equilibrium, balance the masses to show: Substituting mass values and simplifying leads to:$$

a = \frac{1}{10}g.

$$$$

To sketch the velocity-time graph for the motion of stone B, we consider the following points:

1. Initial Condition: At time ( t = 0 ), stone A is released from rest. Hence, the initial velocity of stone B is ( 0 , m/s ).

2. Acceleration: As A descends, B moves up, so it also accelerates but in the opposite direction. The acceleration of B can be derived from acceleration of A using the relation:

$v = u + at$

Since we have already established that the acceleration of A is ( \frac{1}{10}g ), it follows that B accelerates upwards at the same rate.

3. Velocity Increase: As time increases, the velocity of B will continuously increase until it reaches the pulley, at which point it will have a maximum velocity just before the string goes taut.

4. Graph Characteristics: The graph will start from the origin (0,0), and gradually rise linearly with time to indicate uniform acceleration. The slope of the line represents the acceleration of B.

The resulting graph will be a straight line starting from the origin and increasing positively, representing increasing velocity over time until B reaches the pulley.

5. Label: The x-axis will represent time (t), and the y-axis will represent velocity (v).

Graph Description:

• The graph starts at (0,0) and slopes upwards, indicating increasing velocity with time.
• The line is straight, reflecting constant acceleration.
• It will not reach a maximum point on the graph as it represents the moment just before reaching the pulley.

In part (b), we determined the acceleration of A to be ( \frac{1}{10}g ). If we consider the velocity-time graph of B, the working of part (b) is affected by the fact that while A accelerates downwards, this would lead to a downward tension in the string affecting how B accelerates upwards.

This change indicates that the acceleration of A influences both A and B’s motions. Thus, any adjustment in the value of A's acceleration will directly reflect in B's upward velocity and, therefore, in calculations concerning their system dynamics. In essence, the interdependence of their motions underscores how variations in A's dynamics, as indicated by the graph, can impact the computations for B’s acceleration established in part (b).