Two pulleys, 6 in and 2 feet in diameter are connected by a belt. Larger pulley revolves at the rate of 60rev per minute. (a) Find the linear vel of the belt in ft/min (b) Calculate the angular vel of the smaller pulley in rad/min
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(a) The tangential velocity of a pulley is v = r * w = r * (2 * pi * f), where w is the
angular velocity and f is the frequency. The linear velocity of the belt is the same
as the tangential velocity of a pulley. We are given that diameter d = 2 feet for the
larger pulley, so r = d/2 = 1 foot, and that f = 60 rev/min. Thus the linear velocity
of the belt is v = 1 * 2 * pi * 60 = 120*pi feet/min or about 370.0 feet/min.
(b) The tangential velocity of the smaller pulley is the same as the linear velocity
of the belt. With the radius of the smaller pulley being 6/2 = 3 inches or 0.25 feet
we have its angular velocity being w = v / r = 370.0 / 0.25 = 1508 radians/min.
angular velocity and f is the frequency. The linear velocity of the belt is the same
as the tangential velocity of a pulley. We are given that diameter d = 2 feet for the
larger pulley, so r = d/2 = 1 foot, and that f = 60 rev/min. Thus the linear velocity
of the belt is v = 1 * 2 * pi * 60 = 120*pi feet/min or about 370.0 feet/min.
(b) The tangential velocity of the smaller pulley is the same as the linear velocity
of the belt. With the radius of the smaller pulley being 6/2 = 3 inches or 0.25 feet
we have its angular velocity being w = v / r = 370.0 / 0.25 = 1508 radians/min.
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Pulley 6" dia. has radius r=3"
Pulley 2ft dia. = 24" dia. has radius r=12"
Distance along a circle is denoted by (s)
A particle moving arround a circle denoted by velocity (v)
v = s / t from rate = distance/time
s= rΘ from formula of Length of a circular arc
now we have s= (rΘ) / t also written as r (Θ / t)
(Θ / t ) = called angular velocity denoted by (w) (greek letter omega)
thus Linear velocity is v = r w
1 rev = 2π radians
60 rev/min = 120 π radians/min
a particle moving about a circle of radius (r=12") with an angular velocity of 120π radians /min
has a linear velocity v= r w = 12 in * 120 π rad/min = 1440 π in/min
if you think of that particle moving around the pully as being on the belt then
you can see that the belt has a linear velocity the same as the particle.
Belts linear velocity is 1440 π in/min
Angular velocity of smaller pully
if the belt is moving at the linear velocity of 1440 π in/min
then the smaller pulley is also has a linear velocity of 1440π in/min
but the smaller pulley has a radius of ( r = 3" )
if ( v = r w ) then (w = v / r) rearanging the formula
w = ( 1440 π / 3) = 480 π rad/min
Angular velocity of smaller pulley is 480 π rad/min
since 1 rev/min = 2π rad/min then the rotation in rev /min is 240 rev / min
Pulley 2ft dia. = 24" dia. has radius r=12"
Distance along a circle is denoted by (s)
A particle moving arround a circle denoted by velocity (v)
v = s / t from rate = distance/time
s= rΘ from formula of Length of a circular arc
now we have s= (rΘ) / t also written as r (Θ / t)
(Θ / t ) = called angular velocity denoted by (w) (greek letter omega)
thus Linear velocity is v = r w
1 rev = 2π radians
60 rev/min = 120 π radians/min
a particle moving about a circle of radius (r=12") with an angular velocity of 120π radians /min
has a linear velocity v= r w = 12 in * 120 π rad/min = 1440 π in/min
if you think of that particle moving around the pully as being on the belt then
you can see that the belt has a linear velocity the same as the particle.
Belts linear velocity is 1440 π in/min
Angular velocity of smaller pully
if the belt is moving at the linear velocity of 1440 π in/min
then the smaller pulley is also has a linear velocity of 1440π in/min
but the smaller pulley has a radius of ( r = 3" )
if ( v = r w ) then (w = v / r) rearanging the formula
w = ( 1440 π / 3) = 480 π rad/min
Angular velocity of smaller pulley is 480 π rad/min
since 1 rev/min = 2π rad/min then the rotation in rev /min is 240 rev / min