A wheel is turning at a speed of 5 revolutions per second and is then allowed to come to rest. If it does so in 30s , how far did it turn in this process ? Assume uniform acceleration.
Given data is,
| Initial angular velocity ω0 | = | 5 rev/s |
| final angular velocity ω | = | 0 |
| time t | = | 30 s |
| angular displacement θ | = | ? |
| Average angular velocity ω | = | 
|
| = | 
|
|
| = | 2 rev/s | |
| Now angular displacement θ | = | ω t |
| = | 5 rev/s × 30 sec | |
| Since 1 rev | = | 2π radians |
| = | 360 | |
| θ | = | 10π × 30 |
| = | 300 π radians or 54000° | |
| ∴ The wheel turned 54000° in this process. |
The blades of an electric blender are whirling with an angular velocity of 400 rad/s while the first button is pushed in. when the second button is pushed in, the blades accelerate and reach a greater angular velocity after the blades have rotated through an angular displacement of 45 rad. The angular acceleration has a constant value of 1740 rad/s2. Find the final angular velocity of the blades.
| Initial angular velocity ω0 | = | 400 rad/s |
| angular displacement θ | = | 45 rad |
| angular acceleration α | = | 1740 rad/s2 |
| final angular velocity ω | = | ? |
| we know that | ||
| ω2 - ω02 | = | 2α θ |
| ω2 | = | ω02 + 2α θ |
| ω | = | 
|
| = | 
|
|
| = | 563 rad/s | |
| ∴ The final angular velocity is 563 rad/s |
Starting from rest, a fan takes 5 seconds to attain the maximum speed of 400 r.p.m. Assuming constant acceleration, find the time taken by the fan in attaining half the maximum speed.
| Initial angular velocity ω0 | = | 0 |
| Maximum angular velocity ωm | = | 400 rpm |
| = | 
|
|
| = | 
|
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| Time taken to attain maximum velocity t = 5 sec | ||
| constant acceleration α | = | 
|
| = | 
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|
| = | 
|
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| To find the time taken to attain half the maximum speed, | ||
| we use the equation | ||
| ω | = | ω0 + αt |
| here ω | = | 
|
| ω0 | = | 0 |
| α | = |
|
| Substituting these values we get t | ||
| t | = | 
|
| = | 
|
|
| = | 
|
|
| = | 
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| = | 2.5 sec |