Question 38
A uniform solid cuboid
of concrete of dimensions 0.50 m × 1.20
m × 0.40 m and weight 4000 N rests on a flat
surface with the 1.20 m edge vertical as shown in diagram 1.
What is the minimum
energy required to roll the cuboid through 90° to
the position shown in diagram 2 with the 0.50 m edge vertical?
A 200 J B 400 J C 1400 J D 2600 J
Reference: Past Exam Paper – June 2014 Paper 13 Q14
Solution:
Answer:
A.
A force should be provided
such that in the process, only one edge is in contact with the surface (in our
case, this is the bottom right edge). This edge would act as a pivot about
which the cuboid would roll. This is illustrated in the diagrams below.
Since the solid cuboid
is uniform, its centre of mass may be considered to be at the centre of the
cuboid, that is, initially at a height of 0.6 m.
Weight = mg = 4000N
Therefore before the force
is applied,
Potential energy of cuboid
= mgh = mg × h = (4000 × 0.6) J
As illustrated by the
diagram (in the right), as the cuboid rolls about the edge (which is in contact
with the surface), the centre of mass of the cuboid rises to a greater height.
So, its potential energy increases.
The centre of mass rises
to a maximum height when the top left edge is vertically above. So, the potential
energy of the cuboid is maximum at that position. {Afterwards, the cuboid would fall under gravity. So, the
height of its centre of mass decreases.}
Therefore, the minimum
energy produced by the force should cause the potential energy of the cuboid to
change from its original position to the position described above. That is
the minimum energy is equal to the change in energy from these 2 positions.
Consider the position for maximum potential energy. The centre of
mass would be at the middle of the (blue) dotted line. It can be seen that
a right angle triangle is formed with the vertical (blue) dotted line as
the hypotenuse and the other sides being 0.5 m and 1.2 m.
Using Pythagoras’ theorem.
Vertical (blue) dotted line = √(0.52+1.22)
= 1.3 m
Height of centre of mass in new position = 1.3 / 2 = 0.65 m
Potential energy of cuboid
in new position = mgh = (4000 × 0.65) J
Minimum energy required to
roll cuboid = Rise in potential energy
Minimum energy = (4000 × 0.65) – (4000 × 0.60) J
Minimum energy = (4000) × (0.65 – 0.6) = 200 J
Your explanations are just great!!!
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