Sir Lost-a-Lot dons his armor and bursts out
of the castle on his trusty steed Tripper in
his quest to rescue fair damsels from dragons. Unfortunately his valiant aide Doubtless
lowered the drawbridge too far and finally
stopped it 28
◦
below the horizontal. Losta-Lot screeches to a stop when his and his
steed’s combined mass (1400 kg) is 1.2 mmoreno (lm35839) – Homework 15A – smith – (14122) 4
from the end of the bridge, which is 8.5 m
long with a mass of 1100 kg. The lift cable is
attached to the bridge 3.3 m from the hinge
and to the parapets 17 m above the bridge.
α
l x
L
h
Find the tension in the cable. The acceleration of gravity is 9.8 m/s
2
.
Answer in units of kN
To find the tension in the cable, we need to consider the forces acting on the bridge and the equilibrium condition.
Let's start by identifying the forces acting on the bridge. There are three main forces to consider:
1. The weight of the bridge: This is a downward force due to gravity and can be calculated by multiplying the mass of the bridge by the acceleration of gravity. Given that the bridge has a mass of 1100 kg and the acceleration of gravity is 9.8 m/s^2, the weight of the bridge is:
Weight_bridge = mass_bridge * acceleration_gravity
Weight_bridge = 1100 kg * 9.8 m/s^2
2. The weight of Lost-a-Lot and his steed: This is also a downward force due to gravity. Given that their combined mass is 1400 kg, their weight can be calculated similar to the weight of the bridge:
Weight_LostaLot = mass_LostaLot * acceleration_gravity
Weight_LostaLot = 1400 kg * 9.8 m/s^2
3. The tension in the cable: This is an upward force provided by the cable.
Now, let's consider the equilibrium condition. In order for the bridge to be in equilibrium, the sum of the forces in the vertical direction must be zero.
Sum_of_forces_vertical = Weight_bridge + Weight_LostaLot - Tension_cable = 0
Rearranging the equation, we can solve for the tension in the cable:
Tension_cable = Weight_bridge + Weight_LostaLot
Substituting the values we calculated earlier:
Tension_cable = (1100 kg * 9.8 m/s^2) + (1400 kg * 9.8 m/s^2)
Now we can calculate the tension in the cable:
Tension_cable = (1100 * 9.8) + (1400 * 9.8) N
To convert the answer to kilonewtons (kN), we divide the result by 1000:
Tension_cable = ((1100 * 9.8) + (1400 * 9.8)) / 1000 kN
Now you can calculate the tension in the cable by plugging these values into a calculator or doing the multiplication and addition manually.