POWER:
Power is the rate of doing
work.
Power
= Work
done
Time
taken
= Force x
distance
time
= Force x
velocity
P
= fv Units Watts
(W)
Examples:
1.
A man of mass 80 kg climbs a
staircase of height 5 cm in 10s.
Find this average rate of work against gravity.
2.
If the effective power
developed by a crane is 15 kW, find the
time taken to lift a load of weight 50 kN through a vertical of 12
m.
3.
Find the minimum power
needed to pull an object of weight 500 N distance of 8 m up a smooth slope
inclined at 30o to the horizontal in 5 seconds.
4.
A lift of mass 1500 kg is
being raised by an engine working at the rate of 24.5 kW. If the lift rises 50 m in 40 s, find the
load it is carrying.
5.
Water from a river, flowing at ms-1, completely
fills a pipe of cross-sectional area 3 m2. If, when the water is used to drive a
generator, 30% of its kinetic energy is converted into electrical energy, find
the power supplied by the generator.
6.
A pump raises 40 kg of water
a second through a vertical distance of 10 m and delivers it in a jet with speed
12 ms-1. Find the
effective power developed by the pump.
7.
A pump raises water at the
rate of 250 litres per minutes through a vertical distance of 8 m and project it
into a lake at 2 ms-1.
Find the power developed by the pump.
8.
If the engine of a train
travelling at a steady speed of 25 ms-1 is working at a rate of 1800
kW, find the magnitude of the resistance to motion.
9.
A motor-cyclist is riding
along a horizontal road at a
constant speed of 15 ms-1 against a resistance of 80 N. Find the power at which the motor-cycle
engine is working. Given that the
total mass of the rider and his machine is 200 kg, find the initial acceleration
when the power is increased to 1.5 kW.
10. A car of mass 900 kg is
travelling at 20 ms-1
along a horizontal road against a constant resistance of 600 N. Find the power developed by the engine
if the car is (a) moving with constant speed. (b) accelerating at 0.15 ms-2,
(c) decelerating at 0.5 ms-2.
SOLUTIONS:
1. Work done = (80 g x 5)J
P =
80 g x 5
= 392
W
10
2. Power =
15 x 103 W
Load = 50 x 103
N
Distance travelled =
12 m.
Power = f x d
T
Time = f x d
Power
= 50
x 10 x 12
15 x
10
= 40
seconds.
3. Weight – 500
N
Distance = 8 m
Time in 5
seconds.
500 N
Work done =
500 sin 30o x 8 = 400
W
2
Power
=
wd
=
400 = 400
W
T
1
Alternatively.
Power
=
wd
=
500 x h
T
5
H
= 8 sin
30o
Power
=
500 x 8 sin 30o = 400
W
5
4.
Mass of lift = 1500
kg
- (1500 x
10)N.
Power = 24.5
x 103W.
Distance 50 m in time
= 40
seconds load it is carrying = ? = b
kg.
Total mass = (1500 + b)
kg.
Total weight = (1500 +
b)g.
Power = f
x d
T
24.5 x
103 =
10(1500 + b) x 50
40
24.5x103
= 25(1500 + b)
2
49 x
103 = 25(1500 + b)
49 x 103 = 37500
+ 25b
25b = 49000 – 37500
b = 11500
25
b = 460 kg. (Using g
= 10
m/s2)
5. Velocity - 4 m/s
Cross sectional area
= 3
m2
30% of its kinetic energy is
converted into electrical energy, power supplied = ?
Volume per second = (3 x 4)
= 12
m3
of water
=
1000 kg /m3
Mass of water delivered per
second = 12 x 1000 =
12000 kg
At velocity =
4 m/s
k.l of water delivered per second = 1 x 12000 x 4 x 4 = 96000 J
2
Power of generator = 30 x
96000
100
= 28800
W.
=
28.8 kW
6. Mass of water – 40 kg. Per second is raised through distance =
10 m
Speed
= 12
m/s
Effective power
developed. Gain in potential
energy
=
mgh.
= 40 x 10
x 10
= 4000
J
Gain in K.E. = 1 x 40 x 12 x
12
2
= 2880
J
Total energy
= (4000
+ 2880)J
=
6880 J
Power
=
Wd
Time taken
=
6880
1
= 6880
W = 6.88 kW
7. Rate of 250 litres per
minute.
V/min = 250
litres
V/second
=
250 =
25 litres
60
6
distance of 8 m.
Velocity =
2 m/s
Power
developed by the pump.
Density of
water = 1000
kg/cm3
= 1
g/cm3
1000 g - 1000 cm3 -
1l
1 kg - 1 litre
25 l
- 25
kg
6
6
Mass/sec
=
25 kg
6
Gain in PE =
mgh
= 25 x 10 x 8 = 1000
6
3
Gain in kE = 1 mv2 =
1 x 25 x
2 x 2
2 2 6
=
25
3
Total change in =
1000 +
25
Energy 3 3
=
1025
J
3
Power = Wd =
1025 x
1
T 3 1
= 1025
3
(Using g =
10 m/s2) =
341.67
W
8. Speed = 25 m/s
P = 1800 x 103W
At constant speed F =
R
Power
=
FV
P
=
RV
R
= P
= 1800 x 103 = 7.2
V
25
9. Power
= Resistance x
Velocity
= (80 x
15)W
=
1200W
If
total mass of rider + machine is 200 kg.
Initial accn when
power is increased to 1.5 kw.
= 1.5 x 103
W
Initially tractive
force F = P = 1500 =
100N
V
15
200 kg 100
N
80
N
