M1 January 2006 Q5
5.
A parcel of weight 10 N lies on a rough plane inclined at an angle of 30\(^\circ\) to the horizontal. A horizontal force of magnitude \(P\) newtons acts on the parcel, as shown in Figure 2. The parcel is in equilibrium and on the point of slipping up the plane. The normal reaction of the plane on the parcel is 18 N. The coefficient of friction between the parcel and the plane is \(\mu\). Find
(a) the value of \(P\), (4)
(b) the value of \(\mu\). (5)
The horizontal force is removed.
(c) Determine whether or not the parcel moves. (5)
| Scheme | Marks |
|---|---|
| R(perp to plane): \(P\sin 30 + 10\cos 30 = 18\) | M1 A1 |
| Solve: \(P \approx 18.7\) N | M1 A1 |
| (4) |
| Scheme | Marks |
|---|---|
| R(// plane): \(P\cos 30 = 10\sin 30 + F\) | M1 A1 |
| \(F = 18\mu\) used | M1 |
| Sub and solve: \(\mu = 0.621\) or 0.62 | M1 A1 |
| (5) |
| Scheme | Marks |
|---|---|
| Normal reaction now \(= 10\cos 30\) | M1 A1 |
| Component of weight down plane \(= 10\sin 30\ \ (= 5\text{ N})\) (seen) | B1 |
| \(F_{\max} = \mu R_{\text{new}} \approx 5.37\) N (AWRT 5.4) | M1 |
| \(5.37 \gt 5 \;\Rightarrow\;\) does not slide | A1 cso |
| (5) | |
| (14 marks) |