EXPERIMENT NO. 10
Aim: To determine the Mechanical Advantage, Velocity Ratio, & Efficiency of a
Differential Wheel and axle.
Appratus Required:
Differential Wheel and Axle apparatus mounted on a wall, weights hanger, weights, weights
box, 2 rope pieces, string, scale, vernier calipers etc
Theory
1. Simple Wheel and Axle
In fig10.1 is shown a Simple wheel & axle, in which the wheel A & axle B are keyed
to the same shaft. The shaft is mounted on ball bearings, in order to reduce the frictional
resistance to minimum. A string is wound round the axle B, which carries the load to be
lifted. A second string is wound round the wheel A in the opposite direction to that of
string on B.
Let, W = Load lifted
P = Effort applied to lift the load
D = Diameter of effort wheel, and
D = Diameter of the load drum
One end of the string is fixed to the wheel, while the other is free and the effort is
applied to this end. Since the two strings are wound in the opposite directions, so the
downward motion of the effort (P) will raise the load (W).
Since the wheel A and axle B are keyed to the same shaft, so when the wheel rotates
through one revolution, axle will also rotate through one revolution.
We know that the distance moved by the effort in one revolution the effort wheel
= πD
& distance moved by the load in one revolution = πd
V.R. = Distance moved by effort / Distance moved by load = πD/ πd = D/ d
Now ,
M.A. = Load lifted / Effort applied = W/P
and Efficiency
η = M.A. / V.R.
2. Differential Wheel and Axle
It is an improved form of Simple wheel & axle, in which the Velocity Ratio is
intensified with the help of a load axle. In figure 10.2 is shown a Differential wheel &
axle. In this case, load axle BC is made up of two parts of different diameters. Like
Simple wheel & axle, the wheel A, & axle B and C are keyed to the same shaft, which is
mounted on ball bearings, in order to reduce the frictional resistance to minimum.
The effort string is wound round the wheel A. Another string is wound round the axle
B, which after passing round the pulley (to which the weight W is attached) is wound
round the axle C in the opposite directions to that of the axle B; care being taken to wind
the string on the wheel A & axle C in the same direction. As a result of this, when the
string unwinds from the wheel A, the other string also unwinds from the axle C. But it
winds on the axle B in figure 10.2.
Let
D = diameter of effort wheel A,
d1 = diameter of the axle B
d2 = diameter of the axle C
W = Weight lifted by the machine, &
P = Effort applied to lift the weight
We know distance moved by the effort in one revolution of the effort wheel A= πD
Therefore, Length of string, which will wound on the axle B in one revolution = πd1
& Length of string, which will unwound from the axle C in one revolution = πd2
Therefore, Length of string, which will wound in one revolution = πd1 - πd2
= π(d1 - d2)
& distance moved by the weight = 1/2× π (d1 - d2)
= π/2(d1 - d2)
therefore,
and,
V.R. = Distance moved by effort P/ Distance moved by load
= πD/ π/2(d1 - d2)
= 2D/(d1 - d2)
M.A.= W/ P
and, Efficiency,
η = M.A./ V.R.
Observations:
1.
2.
3.
4.
Weight of the empty effort pan P1 =
Weight of the empty hanger, W1 =
Thickness of effort wheel’s string, t1 =
Thickness of load axle’s rope , t2 =
Observation Table:
S.N
O.
Effective circumference of
Wheel
π (D + t1)
1
2
3
Axle
π (d + t2)
Velocity
Ratio
VR =
π (D + t1) /
π (D + t2)
Load
Effort
lifted
Applied
W = W1 + P = P1+P2
W2
Mechanical
Advantage
MA = W/P
Efficiency
= MA/VR
Calculations:
Result:
Average value of: i) M.A. =
ii) V.R. =
iii) % η =
Precautions:
1. Weight of empty hanger and effort pan should always be taken.
2. Weight W should be lifted gradually upwards at uniform speed.
3. To get effective values of circumference, the thickness of the ropes should be
measured.
4. Lubricate the bearings apparatus to reduce friction.
Fig 10.1 Simple Wheel and Axle
Fig 10.2 Wheel and Differential Axle