I\'m new to Work in physics and for some reason I cannot grasp the concept of it

Question

I'm new to Work in physics and for some reason I cannot grasp the concept of it. I have the equations for Work and I know my math up to directional derivatives but I just can't put two and two together when it comes to this topic. I've been scratching my head over this problem for a good hour and have been stuck in an infinite loop.

Can somebody help explain to me how to solve this? The image of the problem is posted below as well as the answer. Hopefully there's a zoom button... this is my first question here on Cramster. I would greatly appreciate any help! Thank you in advance for reviewing my question.

Explanation / Answer

Given data:

length of the bob., L = 2.4 m

the angle makes with vertical, = 35 degree

the initial speed, vi = 8.6 m/s

------------------------------------------

(a)

According to conservation of energy,

total energy (final) = total energy (initial)   

1/2mvf^2 = mgh + 1/2mvi^2

1/2 vf^2 = 1/2vi^2 + g l(1-cos35) ( since h = 1-cos )

0.5*vf^2 = 0.5* 73.96 + 9.8* 2.4*0.1808

vf = 9.081 m/s

(b)

if the bob is in horizontal, h = L

Applying the conservation of energy,

mgL = mgh + 1/2mvi^2

gL = 1/2vi^2 + g l(1-cos35) ( since h = 1-cos )

9.8*2.4 = 0.5* vi^2 + 9.8* 2.4*0.1808

0.5*vi^2 = 23.52 -4.252

Vi = 6.20 m/s

the minimum speed is 6.20 m/s

(c)

if the bob is in same position, h' = l(1-cos180)

Applying the conservation of energy,

mgh' +1/2mv0^2 = mgh + 1/2mgl

gl(1-cos35) + 1/2v0^2 = 1/2gl+ gl(1-cos180) ( since h = 1-cos )

9.8*2.4*0.1808 +1/2v0^2 = 0.5* 23.52 + 47.04

v0 = 10.44 m/s

the minimum speed is 10.44 m/s

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