An electric saw uses a circular spinning blade to slice through wood. When you start the saw, the motor needs 6 seconds of constant angular acceleration to bring the blade to its full angular velocity. If you change the blade so that the rotating portion of the saw now has 5 times its original rotational mass, how long will the motor need to bring the blade to its full angular velocity?

30seconds

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To solve this problem, we need to use the principles of rotational mechanics and the concept of torque.

First, let's use some variables to represent the given information:
- The original rotational mass of the blade is m1.
- The new rotational mass of the blade is m2 = 5m1.
- The time for the motor to bring the blade to its full angular velocity with the original mass is t1 = 6 seconds.
- We need to find the time for the motor to bring the blade to its full angular velocity with the new mass, which we'll call t2.

Now, let's apply the concept of torque, which is the rotational equivalent of force. The torque produced by the motor is given by the equation:

Torque = moment of inertia × angular acceleration

The moment of inertia (I) is a measure of an object's resistance to changes in its rotational motion. For a circular spinning blade, the moment of inertia is given by:

I = 0.5 × mass × radius^2

Since the radius remains the same, we can express the moment of inertia in terms of mass:

I = 0.5 × m × r^2

Now, we can write the torque equation for both cases (initial and final) and set them equal to each other since the torque applied by the motor is the same in both cases:

I1 × angular acceleration1 = I2 × angular acceleration2

Plugging in the expressions for moment of inertia:

(0.5 × m1 × r^2) × angular acceleration1 = (0.5 × m2 × r^2) × angular acceleration2

Mass and radius are constant in this problem, so we can simplify:

m1 × angular acceleration1 = m2 × angular acceleration2

Now, we can relate the angular acceleration to the time using the equation:

Angular acceleration = Change in angular velocity / Time

Since the initial angular velocity is zero and the final angular velocity is full, we can write:

angular acceleration1 = full angular velocity / t1
angular acceleration2 = full angular velocity / t2

Plugging these expressions back into the torque equation:

m1 × (full angular velocity / t1) = m2 × (full angular velocity / t2)

Since the full angular velocity is common to both sides, we can cancel it:

m1 / t1 = m2 / t2

Now, we can solve for t2:

t2 = (m2 / m1) × t1

Substituting the given values:

t2 = (5m1 / m1) × 6

Simplifying:

t2 = 5 × 6
t2 = 30 seconds

Therefore, with 5 times the rotational mass, the motor will need 30 seconds to bring the blade to its full angular velocity.