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If unit of force , velocity &energy are 100 dyne, 10 cm/s & 400 ergs resptivly what will be the unit of mass , length &time . Dimensions of (a, b )in the relation E=(b- x²)/at ,where E is energy,x is dist. and t is time .?
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If unit of force , velocity &energy are 100 dyne, 10 cm/s & 400 ergs r...
Unit Analysis:
- Force unit: 100 dyne
- Velocity unit: 10 cm/s
- Energy unit: 400 ergs
Unit of Mass, Length, and Time:
1. Mass: The unit of force (dyne) is equal to 1 g cm/s². Therefore, 100 dyne = 100 g cm/s². As force = mass x acceleration, we can rearrange this to find the unit of mass: mass = force / acceleration. Given that acceleration has units of cm/s², the unit of mass will be grams (g).
2. Length: The unit of velocity is cm/s. Velocity = distance / time, which means distance = velocity x time. Substituting the given values, we get distance = 10 cm/s x 1 s = 10 cm. Therefore, the unit of length is centimeters (cm).
3. Time: The unit of energy is ergs. Energy = force x distance, so energy = mass x acceleration x distance. Substituting the units we found earlier, ergs = g cm/s² x cm = g cm²/s². Since energy is also defined as power x time, we can rearrange this to find the unit of time: time = energy / power. Given that power has units of erg/s, the unit of time will be seconds (s).
Dimensions of (a, b) in the Relation:
Given the relation E = (b - x²) / at, where E is energy, x is distance, and t is time, we can determine the dimensions of a and b as follows:
- The dimensions of E are ergs (energy), which equals g cm²/s² (as derived earlier).
- The dimensions of x² are cm² (distance squared).
- The dimensions of at are g cm/s² * s = g cm/s (acceleration times time).
- Therefore, the dimensions of (b - x²) are g cm²/s² (energy).
- To make both sides of the equation equal, the dimensions of b should also be g cm²/s², making the dimensions of a equal to 1/s (inverse time).
- Force unit: 100 dyne
- Velocity unit: 10 cm/s
- Energy unit: 400 ergs
Unit of Mass, Length, and Time:
1. Mass: The unit of force (dyne) is equal to 1 g cm/s². Therefore, 100 dyne = 100 g cm/s². As force = mass x acceleration, we can rearrange this to find the unit of mass: mass = force / acceleration. Given that acceleration has units of cm/s², the unit of mass will be grams (g).
2. Length: The unit of velocity is cm/s. Velocity = distance / time, which means distance = velocity x time. Substituting the given values, we get distance = 10 cm/s x 1 s = 10 cm. Therefore, the unit of length is centimeters (cm).
3. Time: The unit of energy is ergs. Energy = force x distance, so energy = mass x acceleration x distance. Substituting the units we found earlier, ergs = g cm/s² x cm = g cm²/s². Since energy is also defined as power x time, we can rearrange this to find the unit of time: time = energy / power. Given that power has units of erg/s, the unit of time will be seconds (s).
Dimensions of (a, b) in the Relation:
Given the relation E = (b - x²) / at, where E is energy, x is distance, and t is time, we can determine the dimensions of a and b as follows:
- The dimensions of E are ergs (energy), which equals g cm²/s² (as derived earlier).
- The dimensions of x² are cm² (distance squared).
- The dimensions of at are g cm/s² * s = g cm/s (acceleration times time).
- Therefore, the dimensions of (b - x²) are g cm²/s² (energy).
- To make both sides of the equation equal, the dimensions of b should also be g cm²/s², making the dimensions of a equal to 1/s (inverse time).
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If unit of force , velocity &energy are 100 dyne, 10 cm/s & 400 ergs resptivly what will be the unit of mass , length &time . Dimensions of (a, b )in the relation E=(b- x²)/at ,where E is energy,x is dist. and t is time .?
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