Please reference the illustration at screenshots<dot>firefox<dot>com/fDDGSaztdbf0b7Uh/brilliant<dot>org
for the following problem:
A bunch of bananas of total weight W is hung at one end of a string passing over a perfectly smooth pulley. At the other end, a monkey starts climbing up the rope at a constant acceleration and covers 16 ft in 2 seconds.
If the bananas always remain at rest, find the weight of the monkey.
Assume that g=32 ft/sec²
Choices:
W/5
2W/5
4W/5
W
5W/4
Thank you.
at the monkeys end:
monkey mass=W/g
tension=(w/g)(g+a)
but d=1/2 a t^2 or a=16*2/4=8 check that
so tension=W/g)(32+8)=40W/32=5W/4
but since tension = bananas
monkey=5W/4
m = mass of monkey
M = mass of bananas
w = weight of monkey
W = weight of bananas
w = m g
W = M g
distance traveled = s
initial velocity = u
acceleration of monkey climbing = a
time = t
SUVAT equations of motion:
s = u t + a t² / 2
16 = 0 ∙ 2 + a ∙ 2² / 2
16 = 0 + a ∙ 4 / 2
16 = 2 a
2 a = 16
a = 16 / 2
a = 8 ft/s²
Since the bananas are stationary, by Newton's third law:
m ( g + a ) = M g
m ( 32 + 8 ) = M ∙ 32
40 m = 32 M
m = 32 M / 40 = 8 ∙ 4 / 8 ∙ 5 = 4 / 5 M
m = ( 4 / 5 ) M
w = W
m g = M g
m g = ( 4 / 5 ) M g
w = ( 4 / 5 ) W
To solve this problem, we can use the concept of Newton's second law of motion. The force acting on an object is equal to its mass multiplied by its acceleration.
In this case, the monkey is climbing up the rope with a constant acceleration. Thus, the upward force applied by the monkey must be equal to the weight of the monkey (since the bananas are at rest and there is no net force on them).
Let's denote the weight of the monkey as M. The acceleration of the monkey is given as 16 ft/2 sec = 8 ft/sec² (since acceleration is the change in velocity divided by the time taken). The acceleration of the monkey is also the acceleration due to gravity (g = 32 ft/sec²).
So, according to Newton's second law, we have:
M * acceleration = M * g
Simplifying the equation, we have:
M * 8 ft/sec² = M * 32 ft/sec²
Dividing both sides by M, we get:
8 ft/sec² = 32 ft/sec²
This equation implies that M cancels out, indicating that the weight of the monkey does not affect the acceleration.
Therefore, the weight of the monkey can be any value, and we cannot determine its weight based on the information given.