A passenger in a train accelerating smoothly away from a station observes that a child’s yo-yo hanging by its string from a luggage rack makes a 10 degree angle from the vertical. Calculate the train’s acceleration?

Calculate the radius of a solid iron sphere that will have a terminal velocity of 100 m/s when dropped from a large height.

Density of iron is 7874 kg/m3. A sphere has Cd = .47. Use p = 1.3kgm3 for air

1.73 m/s²

3.0 cm

Explanation:

Draw a free body diagram of the yo-yo.  There are two forces: weight force mg pulling down, and tension force T pulling up 10° from the vertical.

Sum of forces in the y direction:

∑F = ma

T cos 10° − mg = 0

T cos 10° = mg

T = mg / cos 10°

Sum of forces in the x direction:

∑F = ma

T sin 10° = ma

mg tan 10° = ma

g tan 10° = a

a = 1.73 m/s²

Draw a free body diagram of the sphere.  There are two forces: weight force mg pulling down, and air resistance D pushing up.  At terminal velocity, the acceleration is 0.

Sum of forces in the y direction:

∑F = ma

D − mg = 0

D = mg

½ ρₐ v² C A = ρᵢ V g

½ ρₐ v² C (πr²) = ρᵢ (4/3 πr³) g

3 ρₐ v² C = 8 ρᵢ r g

r = 3 ρₐ v² C / (8 ρᵢ g)

r = 3 (1.3 kg/m³) (100 m/s)² (0.47) / (8 (7874 kg/m³) (9.8 m/s²))

r = 0.030 m

r = 3.0 cm

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