A small package of mass 1.1 kg is held in equilibrium on a rough plane by a horizontal force. The plane is inclined at an angle $\alpha$ to the horizontal, where $\tan \alpha = \frac{3}{4}$. The force acts in a vertical plane containing a line of greatest slope of the plane and has magnitude $P$ newtons, as shown in Figure 2. The coefficient of friction between the package and the plane is 0.5 and the package is modelled as a particle. The package is in equilibrium and on the point of slipping down the plane. (a) Draw, on Figure 2, all the forces acting on the package, showing their directions clearly. (b) (i) Find the magnitude of the normal reaction between the package and the plane. (ii) Find the value of $P$.

A-Level Edexcel Maths Mechanics Further Forces & Newtons Laws: A small package of mass 1.1 kg i

A small package of mass 1.1 kg is held in equilibrium on a rough plane by a horizontal force - Edexcel - A-Level Maths Mechanics - Question 5 - 2009 - Paper 1

Question 5

A small package of mass 1.1 kg is held in equilibrium on a rough plane by a horizontal force. The plane is inclined at an angle $\alpha$ to the horizontal, where $\t... show full transcript

Worked Solution & Example Answer:A small package of mass 1.1 kg is held in equilibrium on a rough plane by a horizontal force - Edexcel - A-Level Maths Mechanics - Question 5 - 2009 - Paper 1

Step 1

Draw, on Figure 2, all the forces acting on the package, showing their directions clearly.

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Answer

In the diagram, the following forces act on the package:

Weight (W): Acts vertically downwards, given by $W = mg = 1.1 imes 9.8 = 10.78$ N.
Normal Reaction (R): Acts perpendicular to the inclined plane.
Applied Force (P): Acts parallel to the plane, directed upwards along the slope.
Frictional Force (F): Acts parallel to the plane, directed downwards along the slope due to the tendency of the package to slip.

Step 2

Find the magnitude of the normal reaction between the package and the plane.

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Answer

Using the equilibrium condition in the vertical direction, we have:

$R = W \cos \alpha + F \sin \alpha$

Since $F = \mu R$ , replacing $F$ gives: $R = 1.1 \cdot 9.8 \cdot \cos \alpha + 0.5R \sin \alpha$

Rearranging gives: $R(1 - 0.5 \sin \alpha) = 10.78 \cos \alpha$

Substituting $\alpha$ to find $R$ :

With $\tan \alpha = \frac{3}{4}$ , calculate: $\cos \alpha = \frac{4}{5}, \sin \alpha = \frac{3}{5}$

Thus, $R = \frac{10.78 \cdot \frac{4}{5}}{1 - 0.5 \cdot \frac{3}{5}} = 9.8 N$

Step 3

Find the value of P.

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Answer

Using the horizontal equilibrium condition:

$P + F \cos \alpha = R \sin \alpha$

Substituting $F = \mu R$ gives: $P + 0.5R \cos \alpha = R \sin \alpha$

Solving for $P$ : $P = R(\sin \alpha - 0.5 \cos \alpha)$

Substituting $R = 9.8$ N, $\sin \alpha = \frac{3}{5}$ , and $\cos \alpha = \frac{4}{5}$ yields: $P = 9.8(\frac{3}{5} - 0.5 \cdot \frac{4}{5}) = 1.96 N$

A small package of mass 1.1 kg is held in equilibrium on a rough plane by a horizontal force - Edexcel - A-Level Maths Mechanics - Question 5 - 2009 - Paper 1

Worked Solution & Example Answer:A small package of mass 1.1 kg is held in equilibrium on a rough plane by a horizontal force - Edexcel - A-Level Maths Mechanics - Question 5 - 2009 - Paper 1

Draw, on Figure 2, all the forces acting on the package, showing their directions clearly.

Find the magnitude of the normal reaction between the package and the plane.

Find the value of P.

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