A box of mass 30 kg is being pulled along rough horizontal ground at a constant speed using a rope - Edexcel - A-Level Maths Mechanics - Question 6 - 2007 - Paper 1

To find the value of P, we need to analyze the forces acting on the box. Given that the box is moving at a constant speed, the net force acting on it must be zero.

1. Identify the forces:

   • Weight of the box: ( W = 30g ) (where ( g \approx 9.81 , ext{m/s}^2 ))
   • Normal force: ( R )
   • Tension in the rope: ( P )
   • Frictional force, which is ( ext{friction} = ext{coefficient} \times R ).

2. Set up the equations:

   For the vertical components: $R + P \sin(20°) = 30g$

   For the horizontal components: $P \cos(20°) = \mu R$, where ( \mu = 0.4 ).

3. Substitute ( R ) from vertical components into horizontal components:

   Rearranging the vertical equation gives: $R = 30g - P \sin(20°)$

   Substituting into the horizontal equation yields: $P \cos(20°) = 0.4(30g - P \sin(20°))$.

4. Solve for P:

   Rearranging gives: $P \cos(20°) + 0.4P \sin(20°) = 0.4 \cdot 30g$

   Factor out ( P ):

   $P(\cos(20°) + 0.4 \sin(20°)) = 0.4 \cdot 30g$.

   Calculate ( g \approx 9.81 , ext{m/s}^2 ):

   Substitute and solve:

   $P = \frac{0.4 \cdot 30 \cdot 9.81}{\cos(20°) + 0.4 \sin(20°)}$.

   On solving, you get ( P \approx 110 , ext{N} ).