You have a grindstone with a mass of 90.0 kg, 0.340-m radius, and is turning at 90.0 rpm. You
press a steel ax against it, bringing it to a stop in 36.0 s. (a) What is the change in kinetic energy of
the grindstone? (b) If the tangential force due to the friction between the ax and the grindstone is
4.00 N, how many rotations will the grindstone make as it comes to a stop?
calculate I of solid disc
omegai = 90 rev/min * 2 pi rad/rev * 1 min/60 s
= 9.42 radians/second
Ke = (1/2) I omegai^2 at start, zero at finish
Torque = 4 R = 4*.34 newton meters
alpha = Torque/I
omega = omegai - alpha t
omega = 0 at finish
so
t = omegai/alpha
which is time to stop
then
d = total radians turned to stop
d = omegai t - .5 alpha t^2
divide that by 2 pi radians/revolution to get number of revolutions
To answer these questions, we need to use the principles of rotational motion and the concepts of work and energy.
(a) To find the change in kinetic energy of the grindstone, we need to calculate its initial and final kinetic energies.
The initial kinetic energy (Ki) can be calculated using the formula: Ki = 0.5 * I * ω₁²
where I is the moment of inertia and ω₁ is the initial angular velocity.
The moment of inertia (I) of a solid disk can be calculated using the formula: I = 0.5 * m * r²
where m is the mass of the grindstone and r is its radius.
Substituting the given values:
I = 0.5 * 90.0 kg * (0.340 m)²
= 5.49 kg m²
Using the conversion factor 2π radians = 1 revolution, we can convert the angular velocity from rpm to rad/s:
ω₁ = 90.0 rpm * (2π rad/1 rev) * (1 min/60 s)
= 9.42 rad/s
Now we can calculate the initial kinetic energy:
Ki = 0.5 * 5.49 kg m² * (9.42 rad/s)²
= 235.38 J
The final kinetic energy (Kf) is zero since the grindstone comes to a stop.
Therefore, the change in kinetic energy (ΔK) is given by: ΔK = Kf - Ki = 0 - 235.38 J = -235.38 J
The change in kinetic energy of the grindstone is -235.38 J.
(b) To find the number of rotations the grindstone makes as it comes to a stop, we need to calculate the work done by the friction force.
The work done (W) can be calculated using the formula: W = ΔK = F * d
where F is the tangential force due to friction and d is the distance traveled.
The distance traveled (d) can be calculated using the formula: d = 2πr * N
where r is the radius of the grindstone and N is the number of rotations it makes.
Rearranging the formula, we find: N = d / (2πr)
The force of friction (F) is given as 4.00 N.
Substituting the given values:
N = (4.00 N * 36.0 s) / (2π * 0.340 m)
= 62.98 rotations
Therefore, the grindstone will make approximately 62.98 rotations as it comes to a stop.