Lizzie is sat securely on a wooden sledge - AQA - A-Level Maths Pure - Question 17 - 2019 - Paper 2

To find the tension $T$ in the rope, we must resolve the forces acting on the sledge in both vertical and horizontal directions.

1. Vertical Forces: The vertical component of the tension must balance the weight of Lizzie and the sledge and the frictional force.

   The equation for vertical forces can be written as: $T \sin \theta - Mg - \mu R = 0 \quad (1)$

   Here, $R$ is the normal reaction force.

2. Horizontal Forces: The horizontal component of the tension must provide the net force causing the acceleration of the system. The net force equation is:

   $T \cos \theta - F_f = Ma \quad (2)$

   where $F_f = \mu R$.

3. Substituting and Combining Equations: From the vertical forces equation (1), we can express $R$ in terms of $T$:

   $R = \frac{T \sin \theta - Mg}{\mu} \quad (3)$

   Substituting equation (3) into equation (2):

   $T \cos \theta - \mu \left( \frac{T \sin \theta - Mg}{\mu} \right) = Ma$

   This leads to: $T \cos \theta - (T \sin \theta - Mg) = Ma$

   Rearranging gives us:

   $T (\cos \theta + \mu \sin \theta) = Ma + Mg$

   Therefore, we find:

   $T = \frac{M(a + \mu g)}{\cos \theta + \mu \sin \theta}$