Power, work and Conservation of Energy discussion

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Power, work and Conservation of Energy discussion
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apradeep92
School: University of Maryland

Please find attached the solutions in word and PDF format. Please note both files are exactly same content-wise. Please let me know if you have any doubt.

Work
1. π‘Šπ‘œπ‘Ÿπ‘˜ π‘‘π‘œπ‘›π‘’ π‘œπ‘› π‘‘β„Žπ‘’ π‘π‘Ÿπ‘Žπ‘‘π‘’ 𝑏𝑦 π‘‘β„Žπ‘’ π‘Žπ‘π‘π‘™π‘–π‘’π‘‘ π‘“π‘œπ‘Ÿπ‘π‘’ = πΉπ‘‘π‘π‘œπ‘ πœƒ
𝐴𝑠 π‘Žπ‘π‘π‘™π‘–π‘’π‘‘ π‘“π‘œπ‘Ÿπ‘π‘’ 𝐹 𝑖𝑠 π‘π‘Žπ‘Ÿπ‘Žπ‘™π‘™π‘’π‘™ π‘‘π‘œ π‘‘π‘–π‘ π‘π‘™π‘Žπ‘π‘’π‘šπ‘’π‘›π‘‘, π‘ π‘œ πœƒ = 0Β°
π‘ π‘œ, π‘Šπ‘œπ‘Ÿπ‘˜ π‘‘π‘œπ‘›π‘’ = πΉπ‘‘π‘π‘œπ‘ 0Β° = 𝑭𝒅
All angels and components are shown in below diagram:

2. 𝐴𝑛𝑔𝑙𝑒 𝑏𝑒𝑑𝑀𝑒𝑒𝑛 π‘”π‘Ÿπ‘Žπ‘£π‘–π‘‘π‘Žπ‘‘π‘–π‘œπ‘›π‘Žπ‘™ π‘“π‘œπ‘Ÿπ‘π‘’ π‘Žπ‘›π‘‘ π‘‘π‘–π‘ π‘π‘™π‘Žπ‘π‘’π‘šπ‘’π‘›π‘‘ = 90Β° + 30Β° = 120Β°
π‘Šπ‘œπ‘Ÿπ‘˜ π‘‘π‘œπ‘›π‘’ π‘œπ‘› π‘‘β„Žπ‘’ π‘π‘Ÿπ‘Žπ‘‘π‘’ 𝑏𝑦 π‘‘β„Žπ‘’ π‘”π‘Ÿπ‘Žπ‘£π‘–π‘‘π‘Žπ‘‘π‘–π‘œπ‘›π‘Žπ‘™ π‘“π‘œπ‘Ÿπ‘π‘’ = 𝐹𝑔 π‘‘π‘π‘œπ‘ 120Β° = 𝐹𝑔 𝑑(βˆ’0.5)
= βˆ’πŸŽ. πŸ“π‘­π’ˆ 𝒅
π‘ π‘œ, π‘Šπ‘œπ‘Ÿπ‘˜ π‘‘π‘œπ‘›π‘’ π‘œπ‘› π‘‘β„Žπ‘’ π‘π‘Ÿπ‘Žπ‘‘π‘’ 𝑏𝑦 π‘‘β„Žπ‘’ π‘”π‘Ÿπ‘Žπ‘£π‘–π‘‘π‘Žπ‘‘π‘–π‘œπ‘›π‘Žπ‘™ π‘“π‘œπ‘Ÿπ‘π‘’ = βˆ’πŸŽ. πŸ“π’Žπ’ˆπ’…
3. 𝐴𝑛𝑔𝑙𝑒 𝑏𝑒𝑑𝑀𝑒𝑒𝑛 π‘›π‘œπ‘Ÿπ‘šπ‘Žπ‘™ π‘“π‘œπ‘Ÿπ‘π‘’ π‘Žπ‘›π‘‘ π‘‘π‘–π‘ π‘π‘™π‘Žπ‘π‘’π‘šπ‘’π‘›π‘‘ = 90Β°
π‘Šπ‘œπ‘Ÿπ‘˜ π‘‘π‘œπ‘›π‘’ π‘œπ‘› π‘‘β„Žπ‘’ π‘π‘Ÿπ‘Žπ‘‘π‘’ 𝑏𝑦 π‘‘β„Žπ‘’ π‘›π‘œπ‘Ÿπ‘šπ‘Žπ‘™ π‘“π‘œπ‘Ÿπ‘π‘’ = 𝐹𝑛 π‘‘π‘π‘œπ‘ 90Β° = 𝐹𝑛 𝑑(0)
π‘ π‘œ, π‘Šπ‘œπ‘Ÿπ‘˜ π‘‘π‘œπ‘›π‘’ π‘œπ‘› π‘‘β„Žπ‘’ π‘π‘Ÿπ‘Žπ‘‘π‘’ 𝑏𝑦 π‘‘β„Žπ‘’ π‘›π‘œπ‘Ÿπ‘šπ‘Žπ‘™ π‘“π‘œπ‘Ÿπ‘π‘’ = 𝟎

4.

𝐴𝑛𝑔𝑙𝑒 𝑏𝑒𝑑𝑀𝑒𝑒𝑛 π‘“π‘Ÿπ‘–π‘π‘‘π‘–π‘œπ‘›π‘Žπ‘™ π‘“π‘œπ‘Ÿπ‘π‘’ π‘Žπ‘›π‘‘ π‘‘π‘–π‘ π‘π‘™π‘Žπ‘π‘’π‘šπ‘’π‘›π‘‘ = 180Β°
π‘Šπ‘œπ‘Ÿπ‘˜ π‘‘π‘œπ‘›π‘’ π‘œπ‘› π‘‘β„Žπ‘’ π‘π‘Ÿπ‘Žπ‘‘π‘’ 𝑏𝑦 π‘‘β„Žπ‘’ π‘“π‘Ÿπ‘–π‘π‘‘π‘–π‘œπ‘›π‘Žπ‘™ π‘“π‘œπ‘Ÿπ‘π‘’ = πΉπ‘˜ π‘‘π‘π‘œπ‘ 180Β° = βˆ’πΉπ‘˜ 𝑑
π‘ π‘œ, π‘Šπ‘œπ‘Ÿπ‘˜ π‘‘π‘œπ‘›π‘’ π‘œπ‘› π‘‘β„Žπ‘’ π‘π‘Ÿπ‘Žπ‘‘π‘’ 𝑏𝑦 π‘‘β„Žπ‘’ π‘“π‘Ÿπ‘–π‘π‘‘π‘–π‘œπ‘›π‘Žπ‘™ π‘“π‘œπ‘Ÿπ‘π‘’ = βˆ’π‘­π’Œ 𝒅

5. As mass doesn’t move perpendicular to ramp, so total force perpendicular to ramp is zero.
so, π‘‡π‘œπ‘‘π‘Žπ‘™ π‘“π‘œπ‘Ÿπ‘π‘’ = πΉπ‘œπ‘Ÿπ‘π‘’ π‘π‘Žπ‘Ÿπ‘Žπ‘™π‘™π‘’π‘™ π‘‘π‘œ π‘Ÿπ‘Žπ‘šπ‘ = 𝐹 βˆ’ πΉπ‘˜ βˆ’ 𝐹𝑔 𝑠𝑖𝑛30Β° = 𝑭 βˆ’ π‘­π’Œ βˆ’ 𝟎. πŸ“π‘­π’ˆ
π‘ π‘œ, π‘‡π‘œπ‘‘π‘Žπ‘™ π‘“π‘œπ‘Ÿπ‘π‘’ = 𝑭 βˆ’ π‘­π’Œ βˆ’ 𝟎. πŸ“π’Žπ’ˆ
6. As
π‘‡π‘œπ‘‘π‘Žπ‘™ π‘“π‘œπ‘Ÿπ‘π‘’ 𝑖𝑠 π‘π‘Žπ‘Ÿπ‘Žπ‘™π‘™π‘’π‘™ π‘‘π‘œ π‘‘π‘–π‘ π‘π‘™π‘Žπ‘π‘’π‘šπ‘’π‘›π‘‘, π‘ π‘œ π‘Šπ‘œπ‘Ÿπ‘˜ π‘‘π‘œπ‘›π‘’ π‘œπ‘› π‘‘β„Žπ‘’ π‘π‘Ÿπ‘Žπ‘‘π‘’ 𝑏𝑦 π‘‘β„Žπ‘’ π‘‘π‘œπ‘‘π‘Žπ‘™ π‘“π‘œπ‘Ÿπ‘π‘’ =
( 𝐹 βˆ’ πΉπ‘˜ βˆ’ 0.5𝐹𝑔 )π‘‘π‘π‘œπ‘ 0Β° = ( 𝐹 βˆ’ πΉπ‘˜ βˆ’ 0.5𝐹𝑔 )𝑑 = ( 𝑭 βˆ’ π‘­π’Œ βˆ’ 𝟎. πŸ“π’Žπ’ˆ)𝒅
We can see that, this is equal to the sum of answers obtained in part 1. To part 4.

Energy
Please note, here, we have assumed that the block is at ground, which is considered as zero
gravitational potential energy level, so 𝑃𝐸 = 𝑃𝐸𝑔 + 𝑃𝐸𝑒 = 0 + 𝑃𝐸𝑒 = 𝑃𝐸𝑒
So, Potential Energy is just the potential ener...

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