OCR MEI M2 2007 January — Question 4 17 marks

Exam BoardOCR MEI
ModuleM2 (Mechanics 2)
Year2007
SessionJanuary
Marks17
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicWork done and energy
TypeMotion on rough inclined plane
DifficultyStandard +0.3 This is a standard M2 friction and energy question with straightforward applications of well-rehearsed techniques: resolving forces on a slope, calculating work done against friction, and using energy conservation. All parts follow predictable patterns with no novel problem-solving required, making it slightly easier than average for A-level mechanics.
Spec3.02h Motion under gravity: vector form3.02i Projectile motion: constant acceleration model3.03t Coefficient of friction: F <= mu*R model3.03u Static equilibrium: on rough surfaces6.02i Conservation of energy: mechanical energy principle6.02j Conservation with elastics: springs and strings6.02m Variable force power: using scalar product
Some tiles on a roof are being replaced. Each tile has a mass of 2 kg and the coefficient of friction between it and the existing roof is 0.75. The roof is at \(30°\) to the horizontal and the bottom of the roof is 6 m above horizontal ground, as shown in Fig. 4. \includegraphics{figure_4}
  1. Calculate the limiting frictional force between a tile and the roof. A tile is placed on the roof. Does it slide? (Your answer should be supported by a calculation.) [5]
  2. The tiles are raised 6 m from the ground, the only work done being against gravity. They are then slid 4 m up the roof and placed at the point A shown in Fig. 4.
    1. Show that each tile gains 156.8 J of gravitational potential energy. [3]
    2. Calculate the work done against friction per tile. [2]
    3. What average power is required to raise 10 tiles per minute from the ground to A? [2]
  3. A tile is kicked from A directly down the roof. When the tile is at B, \(x\) m from the edge of the roof, its speed is \(4 \text{ m s}^{-1}\). It subsequently hits the ground travelling at \(9 \text{ m s}^{-1}\). In the motion of the tile from B to the ground, the work done against sliding and other resistances is 90 J. Use an energy method to find \(x\). [5]
Question 4:
4
3 A box is to be assembled in the shape of the cuboid shown in Fig. 3.1. The lengths are in
centimetres. All the faces are made of the same uniform, rigid and thin material. All coordinates
refer to the axes shown in this figure.
z
G
4 F
y
D E
5
B
C
O 20 A x
Fig. 3.1
(i) The four vertical faces OAED, ABFE, FGCB and CODG are assembled first to make an open
box without a base or a top. Write down the coordinates of the centre of mass of this open box.
[1]
The base OABC is added to the vertical faces.
(ii) Write down the x- and y-coordinates of the centre of mass of the box now. Show that the
z-coordinate is now 1.875. [5]
The top face FGDE is now added. This is a lid hinged to the rest of the box along the line FG. The
lid is open so that it hangs in a vertical plane touching the face FGCB.
(iii) Show that the coordinates of the centre of mass of the box in this situation are (10, 2.4, 2.1).
[6]
[This question is continued on the facing page.]
AnswerMarks
yE
C
Question 4:
4
3 A box is to be assembled in the shape of the cuboid shown in Fig. 3.1. The lengths are in
centimetres. All the faces are made of the same uniform, rigid and thin material. All coordinates
refer to the axes shown in this figure.
z
G
4 F
y
D E
5
B
C
O 20 A x
Fig. 3.1
(i) The four vertical faces OAED, ABFE, FGCB and CODG are assembled first to make an open
box without a base or a top. Write down the coordinates of the centre of mass of this open box.
[1]
The base OABC is added to the vertical faces.
(ii) Write down the x- and y-coordinates of the centre of mass of the box now. Show that the
z-coordinate is now 1.875. [5]
The top face FGDE is now added. This is a lid hinged to the rest of the box along the line FG. The
lid is open so that it hangs in a vertical plane touching the face FGCB.
(iii) Show that the coordinates of the centre of mass of the box in this situation are (10, 2.4, 2.1).
[6]
[This question is continued on the facing page.]
y | E
C
Some tiles on a roof are being replaced. Each tile has a mass of 2 kg and the coefficient of friction between it and the existing roof is 0.75. The roof is at $30°$ to the horizontal and the bottom of the roof is 6 m above horizontal ground, as shown in Fig. 4.

\includegraphics{figure_4}

\begin{enumerate}[label=(\roman*)]
\item Calculate the limiting frictional force between a tile and the roof.

A tile is placed on the roof. Does it slide? (Your answer should be supported by a calculation.) [5]

\item The tiles are raised 6 m from the ground, the only work done being against gravity. They are then slid 4 m up the roof and placed at the point A shown in Fig. 4.

\begin{enumerate}[label=(\Alph*)]
\item Show that each tile gains 156.8 J of gravitational potential energy. [3]

\item Calculate the work done against friction per tile. [2]

\item What average power is required to raise 10 tiles per minute from the ground to A? [2]
\end{enumerate}

\item A tile is kicked from A directly down the roof. When the tile is at B, $x$ m from the edge of the roof, its speed is $4 \text{ m s}^{-1}$. It subsequently hits the ground travelling at $9 \text{ m s}^{-1}$. In the motion of the tile from B to the ground, the work done against sliding and other resistances is 90 J.

Use an energy method to find $x$. [5]
\end{enumerate}

\hfill \mbox{\textit{OCR MEI M2 2007 Q4 [17]}}
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Q1 17 Q2 20 Q3 18 Q4 17