Anna notices a pyramid sticking out of an adjacent ravine and she wants to investigate. She notices some vines hanging over the ravine. To her luck, a strong breeze blows some of the vines toward her and she is able to grab one. The vine appears to make 10° angle with the vertical. She is curious to how long the ravine is so she can make a map of the region later on. (Anna's mass is 48kg.)
___L /|\ L___
(a) Based on the location of the vine, which is right over the middle of the ravine, how wide is the ravine in terms of L?
(b) What she decides to do is tie a rock to the end of the vine and release it like it’s a pendulum. She decides she is going to time the swing to determine the period of the pendulum and from that she could figure out the length of the vine. She counts 10 cycles in 48.17 s. How wide is the ravine? (Hint: Look up the period for a pendulum)
(c) This is it; she has determined the width of the ravine and decides to swing across with out a running start. However, she forgot one thing, centripetal force. It might support her weight, but the vine might not be strong enough to support the swing. What Anna doesn’t know is the vine can only support a tension force of 480 N before it breaks. Will she make it to the other side or fall into the ravine? Prove your answer by finding the maximum tension in the vine.
work:
a) would it be cos(10)=x/L. I'm not sure how to answer this in terms of L.
b) not sure how to attempt. period of pendulum, 1.25?
c) Anna's mass is 48 kg so times that to 9.8 equals 470.4 so the tension must be greater than 470.4N?
Any form of help or guidance will be greatly appreciated. Thank you.
sin 10 = half width/L
so
half width = L sin 10 = .174 L
and'
width = .347 L
or
width/L = .347
----------------------------
only a chicken would look up pendulum period. We are tough.
x = A sin 2pi t/T
v = A(2 pi/T) cos 2 pi t/T
a = -A (2 pi/T)^2 sin 2 pi t/T
or a = -(2 pi/T)^2 x
so
a = - C
F = -m g x/L = m a
so
-m g x/L = - m (2 pi/T)^2 x
or
g/L = (2 pi/T)^2
here we measured T = 4.817 seconds
so
9.81/L = (2 pi/4.18)^2
L = 4.34 meters
so
width = .347 L = 1.51 meters
You are right but we need to find that centripetal acceleration to really answer part c
find max velocity
v = A(2 pi/T) cos 2 pi t/T
max of cos is 1 in the middle of the ravine of course
A = half width of ravine
= 1.51/2 = .755 meter
so vmax = .755(2 pi/4.817)
= .985 meters/second
centripetal =Ac = v^2/r
= .985^2/4.34
= .223 m/s^2
so max tension = ma + mg
= 48( .223 + 9.81)
=482 Newtons
that is a little more than we need to break the vine
Thank you so much Damon! I really appreciate it!! I could not understand any of this without your help!
a) To determine the width of the ravine in terms of L, we can use trigonometry. Since the vine makes a 10° angle with the vertical, we can assume the vine is perpendicular to the ground, and the angle between the vine and the ravine is 80° (90° - 10°). We can define the width of the ravine as x.
Using trigonometry, we can establish the following relationship:
cos(80°) = x/L
Since we need to express the width of the ravine in terms of L, we can rearrange the equation to solve for x:
x = L * cos(80°)
b) To determine the length of the vine, we can use the period formula for a pendulum. The period (T) of a pendulum is defined as the time taken to complete one full swing.
T = 2π√(L/g)
In this case, we know the number of cycles (N) and the time taken (t) to complete those cycles:
N = 10 cycles
t = 48.17 s
The period (T) can be calculated by dividing the total time taken by the number of cycles:
T = t / N
Substituting the values into the formula, we can solve for T:
T = 48.17 s / 10
Once we have the period, we can rearrange the formula for T to solve for L:
L = (T^2 * g) / (4π^2)
With the value of g (acceleration due to gravity) being approximately 9.8 m/s^2, we can substitute the values into the formula to find the length of the vine (L).
c) To determine if the vine can support Anna's weight, we need to find the tension force in the vine when she is swinging across the ravine.
The tension force (T) can be calculated using the formula:
T = (m * v^2) / r
Where:
m = mass of Anna
v = velocity of Anna while swinging
r = radius of the swing (distance between Anna's center of mass and the axis of rotation)
Since Anna is swinging without a running start, we can assume her velocity while swinging is equal to the length of the vine (L) multiplied by the angular velocity (ω).
v = L * ω
The angular velocity (ω) can be calculated using the formula:
ω = 2π / T
Once we have the angular velocity, we can calculate the radius of the swing (r) using the formula:
r = L * sin(10°)
Substituting the values into the formula for tension force (T), we can find the maximum tension in the vine and compare it to the maximum tension the vine can withstand (480 N) to determine if Anna will make it across or fall into the ravine.