Final answer:
To calculate how long it takes for the grindstone to come to rest, the frictional force is found, then the torque caused by this force, and finally the angular deceleration. The grindstone's initial angular velocity is converted from RPM to rad/s and then utilized along with angular deceleration to determine the time until rest.
Step-by-step explanation:
The question is asking us to determine how long it will take for a grindstone to come to rest when an ax is pressed against it. This involves calculations pertaining to angular motion and friction. Given that the grindstone has a mass of 50 kg and a diameter of 0.52 m (radius of 0.26 m), it is rotating at 1000 RPM (revolutions per minute), and there is a normal force of 160 N applied to its rim with a coefficient of friction of 0.8, we can find the torque caused by friction and subsequently use it to calculate the angular deceleration. Finally, we can determine the time it takes for the grindstone to stop rotating.
Firstly, we can calculate the frictional force using the formula: Frictional force = Normal force x coefficient of friction. This gives us a frictional force of 160 N x 0.8 = 128 N. Then, we find the torque by multiplying the frictional force by the radius of the disk (Torque = radius x Frictional force), resulting in a torque of 0.26 m x 128 N = 33.28 Nm. We then use Newton's second law for rotation (Torque = Moment of inertia x angular acceleration) to find the angular acceleration. The moment of inertia for a solid disk is given by I = 0.5 x mass x radius^2, which calculates to 0.5 x 50 kg x (0.26 m)^2 = 1.69 kgm^2. Now, angular acceleration (α) can be calculated as Torque / Moment of inertia, resulting in 33.28 Nm / 1.69 kgm^2 = 19.69 rad/s^2.
To find the time it takes for the grindstone to stop, we calculate the initial angular velocity (ω_initial) by converting 1000 RPM to rad/sec (ω_initial = 1000 revolutions/minute * 2π rad/revolution * 1 minute/60 seconds = 104.72 rad/s). Then, using the formula ω_final = ω_initial + α*t and noting that the final angular velocity (ω_final) will be 0 rad/s when the grindstone comes to rest, we find the time (t) as t = -ω_initial / α, which gives us approximately 5.32 seconds.