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26 November, 02:23

How fast do you need to swing a 250-g ball at the end of a string in a horizontal circle of 0.6-m radius so that the string makes a 39 degree angle relative to the horizontal?

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  1. 26 November, 02:55
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    2.7 m/s

    Explanation:

    Draw a free body diagram of the ball. There are two forces:

    Weight force mg pulling down

    Tension force T pulling 39° above the horizontal

    Sum of the forces in the y direction:

    ∑F = ma

    T sin θ - mg = 0

    T = mg / sin θ

    Sum of the forces in the radial (+x) direction:

    ∑F = ma

    T cos θ = m v² / r

    Substitute:

    (mg / sin θ) cos θ = m v² / r

    mg / tan θ = m v² / r

    g / tan θ = v² / r

    v = √ (gr / tan θ)

    Given that r = 0.6 m and θ = 39°:

    v = √ (9.8 m/s² * 0.6 m / tan 39°)

    v ≈ 2.7 m/s
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