Download answers - Dr. J. Welch

Moments and Equilibrium
Spanners
Moment
=
=
=
Force
50 N
10 Nm
x Distance (at right angles)
x 0.2 m
Moment
Force
=
=
=
=
Force
x Distance (at right angles)
Moment  Distance (at right angles)
10 Nm
 0.25 m
40 N
Moment = 0 Nm if the distance at right angles is zero; therefore make
the force parallel to the spanner handle.
See-saw
Left-hand force causes an anti-clockwise moment.
Right-hand force causes a clockwise moment.
Call the overall clockwise moment M.
M
=
=
=
=
Clockwise moment
( 60N x 2.3m )
13.8 Nm
0.2 Nm
–
–
–
anti-clockwise moment
( 40N x 3.4m )
13.6 Nm
There is a positive clockwise moment of 0.2 Nm, so the see-saw rotates
clockwise (down to the right).
Crane
The pivot is the end corner of the dockside.
Counterweight and crane body cause an anti-clockwise moment.
Load causes a clockwise moment.
Call the overall clockwise moment M.
M
=
=
=
Clockwise moments –
anti-clockwise moment
( 50,000N x 20m ) + ( 5,000N x 4m ) – ( L N x 15m)
1,000,000 Nm
+ 20,000 Nm
– 15L Nm
If the crane is just stable, then M will be zero. So …
15L
= 1,000,000 Nm
+ 20,000 Nm
L
= 1,020,000 Nm
 15 m
= 68,000 N
So the crane is just balanced if lifting a load of 68,000N – any more than
this and it will fall into the docks.
Bus
If the bus is symmetrical, the CofM is 3m up and 1.5m across. The bus
falls if this point is outside of the supports (the wheels).
(You can calculate what angle this is using trigonometry …
3m
tan(angle)
Max angle
= 1.5 / 3
= 26.6º
)
1.5m
It’s better for the passengers to sit on the bottom because this will make
the bus heavier at the bottom so its CofM will be less than 3m up.
Collisions and Momentum
Vectors and Scalars
Scalars: mass, length, time, speed, energy, power, frequency,
temperature, angle …
Vectors: velocity, momentum, acceleration, force …
Cars
(i)
Red car
Momentum =
=
=
Mass
x Velocity
500 kg
x 20 m/s to the right
10,000 kg m/s to the right
Blue car
Momentum =
=
=
Total
=
=
Mass
x Velocity
500 kg
x 20 m/s to the left
10,000 kg m/s to the left
10,000 kg m/s left + 10,000 kg m/s right
0
During the crash, momentum is conserved, so it stays at 0.
After the crash, both cars must have speed 0.
(ii)
Red car
Momentum =
=
=
Mass
x Velocity
500 kg
x 30 m/s to the right
15,000 kg m/s to the right
Blue car
Momentum =
=
=
Mass
x Velocity
500 kg
x 20 m/s to the left
10,000 kg m/s to the left
Total
=
=
10,000 kg m/s left + 15,000 kg m/s right
5,000 kg m/s to the right
After the crash, momentum is still 5,000 kg m/s right.
Wreckage
Momentum =
Velocity
=
=
=
Mass
Momentum
5,000 kg m/s right
5 m/s to the right
Rocket
Before firing
Momentum = Mass
x Velocity
= 1,000 kg x 0 m/s
x


Velocity
Mass
1,000 kg
=
0 kg m/s
After firing it must be the same – zero.
Gas
Momentum =
=
=
Mass
x Velocity
200 kg
x 200 m/s left
40,000 kg m/s left
Rocket
Must be 40,000 kg m/s right so that the total is still zero.
Momentum =
Velocity
=
=
=
Mass
Momentum
40,000 kg m/s right
50 m/s right
x


Velocity
Mass
800 kg
Advanced
The ship will go slower than we worked out because it doesn’t get that
much lighter all in one go. The way to do the sum is a little bit at a time;
taking steps of say 10kg of fuel, all the way up to 200kg used. To be
more accurate, you make the steps smaller, say every 1kg of fuel, but it
takes longer to do. Newton worked out a trick to speed up doing the sum
even with the tiniest of steps. It’s called calculus. The real final speed in
our example would be 44.6 m/s.
Energy – Snooker
Momentum before collision
Red ball
=
White ball =
=
0 kg m/s
100 g
x 1 m/s
0.1 kg m/s
Momentum after collision
Red ball
=
=
100 g
x 0.8 m/s
0.08 kg m/s
White ball =
=
100 g
x 0.2 m/s
0.02 kg m/s
Total
0.1 kg m/s
same as before
=
=
KE before collision
Red ball
=
White ball =
=
=
0J
½ x Mass x Speed2
½ x 100 g x (1 m/s)2
0.05 J
KE after collision
Red ball
=
=
=
½ x Mass x Speed2
½ x 100 g x (0.8 m/s)2
0.032 J
White ball =
=
=
½ x Mass x Speed2
½ x 100 g x (0.2 m/s)2
0.002 J
Total
=
0.034 J
“Lost”
=
=
0.05 – 0.034
0.016 J
“Lost” energy was converted to sound and heat.
Force – Seatbelts
Momentum =
=
=
Mass
x Velocity
50 kg
x 32 m/s forwards
1600 kg m/s forwards
Seatbelt
1600 kg m/s lost in 1s (i.e. 1.6kN)
Windscreen
1600 kg m/s lost in 0.001s so 1,600,000 kg m/s (1600 / 0.001) would be
lost in a whole 1s (i.e. 1.6MN)
Skull
Survives OK with the seatbelt; but busts open on the windscreen. The
airbag makes it take even longer to stop you, so the force is reduced. It’s
nothing to do with the bag being made of a soft material; the bag could
be hard, so long as it has enough ‘give’ in it to slow you down gradually.
Force – Saturn V
Momentum =
=
=
Mass
x Velocity
3 M kg
x 1500 m/s
4500 M kg m/s
Force
Momentum

4500 M kg m/s 
30 MN
=
=
=
Time taken
150 s
It’s only an estimate because this is the average thrust over the 150s;
the thrust was probably not really constant over this time.
Orbits
Moving in a Circle
Experiment
The factors that affect the orbit radius are the force applied to the
string (the weights you attach), the speed of whirling, and the mass of
plasticene. You should find that the speed makes a lot more difference
than the other factors.
For the mathemagically minded, the formula for the force required is …
F
mv 2
r
where m is the mass, v the speed and r the radius. The speed is squared
which is why it makes more difference. You could estimate some values
from your experiment to confirm the formula – it will be only very
approximate.
Question
The table of data is a red herring. The centripetal force is provided by
the Sun’s gravity which is strongest nearest the Sun. Therefore Mercury
must need the biggest force! The mathematicians can work it out using
the formula above; Mercury’s high speed is much more important than
Jupiter’s mass or Pluto’s distance because the speed gets squared.
Notes
Centripetal force in orbits is provided by gravity.
Centripetal force in atoms is provided by electrostatic attraction (a
positive nucleus attracting the negative electrons).
Geophysics
Wegener