Decorative.
Circular Motion
Vertical Circles
With Dr. Trevor Johnson-Steigelman
By the end of this lesson, you will be able…
1.
To draw the free body diagram for objects traveling in vertical circles at various points in the
path.
2.
To solve problems involving objects moving in vertical circles when gravity is involved.
3.
To determine the minimum speed, radius, centripetal force, or centripetal acceleration for
objects moving in vertical circles.
This presentation by
H. Trevor Johnson-Steigelman
is licensed under a
Creative Commons
Attribution-NonCommercialShareAlike 4.0 International License.
Decorative.
Check Yourself
A ball tied to a string is moving in a
vertical circle at a constant speed. How
do the tensions at the top and bottom of
the path compare?
?
𝐹𝑇𝑡𝑜𝑝
1. The tension at the top is larger.
2. The tension at the bottom is larger.
3. The tensions are the same.
Note: Constant speed in this situation is tough,
but not impossible to produce.
?
𝐹𝑇𝑏𝑜𝑡
Decorative.
The Tension in the String Will Change for a Vertical Circle
Assuming a constant velocity
(which is hard to do perfectly),
the centripetal force will be
constant.
This means that the
tension will change
depending on the location.
In a roller coaster, the
normal force would act the
same way.
This is why you feel like
you will fall out at the top,
and feel “crushed” at the
bottom.
Side
Vie
w
𝐹Ԧ𝑇
𝐹Ԧ𝑔
𝑟Ԧ
𝐹Ԧ𝑇
𝐹Ԧ𝑇
𝐹Ԧ𝑔
𝐹Ԧ𝑔
𝐹Ԧ𝑇
𝐹Ԧ𝑔
Decorative.
Vertical Circle:
Top of circle
𝐹Ԧ𝑇
𝐹Ԧ𝑔
+
Bottom of circle
Σ𝐹Ԧ = 𝑚 𝑎Ԧ𝑐
𝐹Ԧ𝑔 + 𝐹Ԧ𝑇 = 𝑚 𝑎Ԧ𝑐
+
𝐹𝑔 + 𝐹𝑇 = 𝑚 𝑎𝑐
same sign:
both in same direction
𝐹𝑇 = 𝑚 𝑎𝑐 − 𝐹𝑔
+ ≡ towards center
Directions: + is in
- is out
Σ𝐹Ԧ = 𝑚 𝑎Ԧ𝑐
𝐹Ԧ𝑇
𝐹Ԧ𝑔
𝐹Ԧ𝑔 + 𝐹Ԧ𝑇 = 𝑚 𝑎Ԧ𝑐
−
+
𝐹𝑔 + 𝐹𝑇 = 𝑚 𝑎𝑐
opposite signs:
point in opposite
directions
𝐹𝑇 = 𝑚 𝑎𝑐 + 𝐹𝑔
Decorative.
Vertical Circles
What is the minimum speed for a person to not fall out of a
car on a roller coaster with no safety harness?
Top of circle
𝐹𝑁
Σ𝐹Ԧ = 𝑚 𝑎Ԧ𝑐
𝐹Ԧ𝑔
𝐹Ԧ𝑔 + 𝐹Ԧ𝑁 = 𝑚 𝑎Ԧ𝑐
+
+
+
𝐹𝑔 + 𝐹𝑁 = 𝑚 𝑎𝑐
same sign:
same
𝑣2
direction
+
𝐹𝑔 + 𝐹𝑁 = 𝑚
𝑟
At minimum
speed, 𝐹𝑁 = 0
+
𝐹𝑔 = 𝑚
𝑚𝑔=𝑚
𝑔=
𝑣2
𝑟
𝑣2
𝑟
𝑣2
𝑟
𝑣 = 𝑔𝑟
Decorative.
Review:
• For uniform circular motion in a vertical circle,
the tension (or the normal force) changes.
• The minimum speed possible occurs when the
tension is zero at the top, so that gravity alone is
providing the normal force.
References:
Paul Peter Urone, Roger Hinrichs, College Physics, OpenStax, Houston, TX, Jun 21, 2012
https://openstax.org/books/college-physics/pages/preface
Samuel J. Ling, William Moebs, Jeff Sanny, University Physics Volume 2, OpenStax, Houston, TX, Oct 6, 2016
https://openstax.org/books/university-physics-volume-2/pages/1-introduction