A cooling fan is turned off when it is running at 850 rev/min. It turns 1500 revolutions before it comes to a stop. (a) What was
1 answer:
Answer
given,
cooling fan revolution = 850 rev/min
fan turns before revolution = 1500 revolutions
θ = 1500 revolution
θ = 1500 x 2 x π
θ = 9424.78 rad
a) using equation of rotation
ω² = ω₀² + 2 α θ
ω = 0 because body comes to rest
0 = 89² + 2 x α x 9424.78
α = -0.42 rad/s²
b) time take for the fan to stop
ω = ω₀ + α t
0 = 89 - 0.42 t
t = 211.9 s
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(a) 1.43 m/s
We can solve this problem by using the law of conservation of energy.
The initial total energy stored in the spring-mass system is
where
k = 7.91 N/m is the spring constant
Substituting,
The final kinetic energy of the ball is equal to the energy released by the spring + the work done by friction:
where
is the kinetic energy of the ball, with
being the mass of the ball
v being the final speed
is the work done by friction (which is negative since the force of friction is opposite to the motion), where
is force of friction
d = 14.5 cm = 0.145 m is the displacement
Substituting,
So, the kinetic energy of the ball as it leaves the cannon is
And so the final speed is
(b) +5.08 cm
The speed of the ball is maximum at the instant when all the elastic potential energy stored in the spring has been released: in fact, after that moment, the spring does no longer release any more energy, so the kinetic energy of the ball from that moment will start to decrease, due to the effect of the work done by friction.
The elastic potential energy of the spring is
And this has all been released when it becomes zero, so when x = 0 (equilibrium position of the spring). However, the spring was initially compressed by 5.08 cm, so the ball has maximum speed when
x = +5.08 cm
with respect to the initial point.
(c) 1.78 m/s
The maximum speed is the speed of the ball at the moment when the kinetic energy is maximum, i.e. when all the elastic potential energy has been released.
As we calculated in part (a), the total energy released by the spring is
E = 0.0102 J
The work done by friction here is just the work done to cover the distance of
d = 5.08 cm = 0.0508 m
Therefore
So, the kinetic energy of the ball at the point of maximum speed is
And so the final speed is