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12 November, 17:28

A carpenter uses a hammer to strike a nail. Approximate the hammer's weight of 1.8lbs, as being concentrated at the head, and assume that at impact the head is traveling in the - j direction. If the hammer contacts the nail at 50mph and the impact occurs over 0.023 seconds, what is the magnitude of the average force exerted by the nail of the hammer?

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  1. 12 November, 18:41
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    The average force F exherted by the nail over the hammer is 178.4 lbf.

    Explanation:

    The force F exherted by the nail over the hammer is defined as:

    F = |I|/Δt

    Where I and Δt are the magnitude of the impact and the period of time respectively. We know that the impact can be calculated as the difference in momentum:

    I = ΔP = Pf - Pi

    Where Pf and Pi are the momentum after and before the impact. Recalling for the definition for momentum:

    P = m. v

    Where m and v are the mass and the velocity of the body respectively. Notice that final hummer's momentum is zero due to the hammers de-acelerate to zero velocity. Then the momentum variation will be expressed as:

    ΔP = - Pi = - m. vi

    The initial velocity is given as 50 mph and we will expressed in ft/s:

    vi = 50 mph * 1.47 ft/s/mph = 73.3 ft/s

    By multiplyng by the mass of 1.8 lbs, we obtain the impulse I:

    |I| = |ΔP| = |-m. vi| = 1.8 lb * 73.3 ft/s = 132 lb. ft/s

    Dividing the impulse by a duration of 0.023 seconds, we finally find the force F:

    F = 132 lb. ft/s / 0.023 s = 5740 lb. ft/s^2

    Expressing in lbf:

    F = 5740 lb. ft/s^2 * 0.031 lbf/lb. ft/s^2 = 178.4 lbf
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