Part of riding a bicycle involves leaning at the correct angle when making a turn, as seen below. To be stable, the force exerted by the ground must be on a line going through the center of gravity. The force on the bicycle wheel can be resolved into two perpendicular components—friction parallel to the road (this must supply the centripetal force ) and the vertical normal force (which must equal the system’s weight). (a) Show that θ (as defined as shown) is related to the speed v and radius of curvature r of the turn in the same way as for an ideally banked roadway—that is, θ = tan − 1 ( v 2 / r g ) . (b) Calculate θ for a 12.0-m/s turn of radius 30.0 m (as in a race).
Part of riding a bicycle involves leaning at the correct angle when making a turn, as seen below. To be stable, the force exerted by the ground must be on a line going through the center of gravity. The force on the bicycle wheel can be resolved into two perpendicular components—friction parallel to the road (this must supply the centripetal force ) and the vertical normal force (which must equal the system’s weight). (a) Show that θ (as defined as shown) is related to the speed v and radius of curvature r of the turn in the same way as for an ideally banked roadway—that is, θ = tan − 1 ( v 2 / r g ) . (b) Calculate θ for a 12.0-m/s turn of radius 30.0 m (as in a race).
Part of riding a bicycle involves leaning at the correct angle when making a turn, as seen below. To be stable, the force exerted by the ground must be on a line going through the center of gravity. The force on the bicycle wheel can be resolved into two perpendicular components—friction parallel to the road (this must supply the centripetal force) and the vertical normal force (which must equal the system’s weight). (a) Show that
θ
(as defined as shown) is related to the speed
v
and radius of curvature
r
of the turn in the same way as for an ideally banked roadway—that is,
θ
=
tan
−
1
(
v
2
/
r
g
)
. (b) Calculate
θ
for a 12.0-m/s turn of radius 30.0 m (as in a race).
Definition Definition Force on a body along the radial direction. Centripetal force is responsible for the circular motion of a body. The magnitude of centripetal force is given by F C = m v 2 r m = mass of the body in the circular motion v = tangential velocity of the body r = radius of the circular path
Part of riding a bicycle involves leaning at the correct angle when making a turn, as seen. To be stable, the force exerted by the ground must be on a line going through the center of gravity. The force on the bicycle wheel can be resolved into two perpendicular components—friction parallel to the road (this must supply the centripetal force), and the vertical normal force (which must equal the system’s weight).
(a) Show that θ (as defined in the figure) is related to the speed v and radius of curvature r of the turn in the same way as for an ideally banked roadway—that is, θ = tan–1v2/ rg
(b) Calculate θ for a 12.0 m/s turn of radius 30.0 m (as in a race).
Consider a ball rolling around in a circular path on the inner surface of a cone. The weight of the ball is shown by the vector W. Without friction, only one other force acts on the ball—a normal force, (a) Draw in the vector for the normal force. (The length of the vector depends on the next step, b.) (b) Using the parallelogram rule, show that the resultant of the two vectors is along the radial direction of the ball’s circular path.
Consider a ball rolling around in a circular path on the inner surface of a cone. The weight of the ball is shown by the vector W. Without
friction, only one other force acts on the ball-a normal force, (a) Draw in the vector for the normal force. (The length of the vector depends
on the next step, b.) (b) Using the parallelogram rule, show that the resultant of the two vectors is along the radial direction of the ball's
circular path. (Yes, the normal is appreciably larger than the weight!)
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