1. Horizontal and Vertical Circles
Car on Curve
Car on banked Curve
Horizontal vs. Vertical Circle
Vertical circle at top
Examples

2. More complicated examples 1
Car on Track with Friction
Example 5-6
μ mg = mv2/r

3. More complicated examples 2
Car on Banked Track
Example 5-7
Don’t use tilted axes
FN cosθ = mg,
FN sinθ = mv2/r
tanθ = v2/rg

4. Horizontal vs. Vertical Circle
Horizontal - circular perp. (┴) to gravity
Vertical – circular inline with gravity
Situations
Ball on a cord (tension only pulls in)
Car going over hump (normal pushes up)

5. Vertical circle at the top
“Inside” Problem – Ball on string.
Starts fast, slows down
mg + FT = mv2/r
“Outside Problem” – Car on hump.
Starts slow, speeds up
mg - FN = mv2/r mg FT
Need both mg and FT at high speeds. As RHS gets smaller, mg takes over and FT not needed
Required ma starts high, gets smaller as v decreases mg FN
FN balances mg at low speeds. As RHS gets larger, more mg required and FN not needed
Required ma starts low, gets larger as v increases

6. Vertical Circle Examples
Situations
Ball on a cord (tension only pulls in)
Car going over hump (normal pushes up)
Example 5-4 (ball on vertical string)
(“inside problem”)
Problem 14 (sports car on hump)
(“outside problem”)
Problem 15 (ferris wheel)
Problem 16 (bucket with water)

7. Examples - “Inside” Vertical Circle
Example 5-4 (ball on vertical string)
0.15 kg ball on 1.1 m cord
What speed does string go slack at top?
𝐹 𝑇 +𝑚𝑔= 𝑚 𝑣 2 𝑟 𝑣= 𝑟𝑔
What is tension for v = 5 m/s?
Problem 16 (bucket with water)
2 kg bucket on 1.1 m cord
Tension at bottom 25 N, what is speed at bottom?
𝐹 𝑇 −𝑚𝑔= 𝑚 𝑣 2 𝑟 𝑣= 𝑟 𝐹 𝑇 −𝑚𝑔 𝑚 = 𝑚 25𝑁−19.6𝑁 2 𝑘𝑔 =1.72 𝑚/𝑠
𝐹 𝑇 +𝑚𝑔= 𝑚 𝑣 2 𝑟 𝑣= 𝑟𝑔 =3.3 𝑚/𝑠

8. Examples - “Outside” Vertical Circle
Problem 14 (sports car on hump)
950 kg car, 95 m radius hump
Goes over top at 22 m/s
What is normal force of road?
𝑚𝑔−𝐹 𝑁 = 𝑚 𝑣 2 𝑟 𝐹 𝑁 =𝑚𝑔− 𝑚 𝑣 2 𝑟
What is normal force on 72 kg driver?
What speed does driver feel weightless?
𝑚𝑔−𝐹 𝑁 = 𝑚 𝑣 2 𝑟 𝑣= 𝑟𝑔
Problem 15 (ferris wheel)
7.5 m radius ferris wheel
weightless at top
𝑚𝑔−𝐹 𝑁 = 𝑚 𝑣 2 𝑟 𝑣= 𝑟𝑔 =8.6 m/s
𝑓= 𝑣 2𝜋𝑟 =0.18 𝑟𝑒𝑣 𝑠 =10.9 𝑟𝑒𝑣 𝑚𝑖𝑛

9. Example – Roto-Ride 4.6 m drum radius
0.5 rev/sec frequency (14.45 m/s)
Radial
𝐹 𝑁 = 𝑚 𝑣 2 𝑟
Vertical
𝐹 𝑓𝑟 =𝑚𝑔=0
𝜇 𝐹 𝑁 =𝑚𝑔
Dividing vertical by Radial
𝜇= 𝜇 𝐹 𝑁 𝐹 𝑁 = 𝑚𝑔 𝑚 𝑣 2 𝑟 = 𝑟𝑔 𝑣 2 =0.22