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28 October, 10:55

Now let's apply the work-energy theorem to a more complex, multistep problem. In a pile driver, a steel hammerhead with mass 200 kg is lifted 3.00 m above the top of a vertical I-beam that is to be driven into the ground (Figure 1). The hammer is then dropped, driving the I-beam 7.40 cm farther into the ground. The vertical rails that guide the hammerhead exert a constant 60.0 N friction force on it. Use the work-energy theorem to find

(a) the speed of the hammerhead just as it hits the I-beam and

(b) the average force the hammerhead exerts on the I-beam.

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  1. 28 October, 13:52
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    a) v = 7.67

    b) n = 81562 N

    Explanation:

    Given:-

    - The mass of hammer-head, m = 200 kg

    - The height at from which hammer head drops, s12 = 3.00 m

    - The amount of distance the I-beam is hammered, s23 = 7.40 cm

    - The resistive force by contact of hammer-head and I-beam, F = 60.0 N

    Find:-

    (a) the speed of the hammerhead just as it hits the I-beam and

    (b) the average force the hammerhead exerts on the I-beam.

    Solution:-

    - We will consider the hammer head as our system and apply the conservation of energy principle because during the journey of hammer-head up till just before it hits the I-beam there are no external forces acting on the system:

    ΔK. E = ΔP. E

    K_2 - K_1 = P_1 - P_2

    Where, K_2: Kinetic energy of hammer head as it hits the I-beam

    K_1: Initial kinetic energy of hammer head ( = 0) ... rest

    P_2: Gravitational potential energy of hammer head as it hits the I-beam. (Datum = 0)

    P_1: Initial gravitational potential energy of hammer head

    - The expression simplifies to:

    K_2 = P_1

    Where, 0.5*m*v2^2 = m*g*s12

    v2 = √ (2*g*s12) = √ (2*9.81*3)

    v2 = 7.67 m/s

    - For the complete journey we see that there are fictitious force due to contact between hammer-head and I-beam the system is no longer conserved. All the kinetic energy is used to drive the I-beam down by distance s23. We will apply work energy principle on the system:

    Wnet = (P_3 - P_1) + W_friction

    Wnet = m*g*s13 + F*s23

    n*s23 = m*g*s13 + F*s23

    Where, n: average force the hammerhead exerts on the I-beam.

    s13 = s12 + s23

    Hence,

    n = m*g * (s12/s23 + 1) + F

    n = 200*9.81 * (3/0.074 + 1) + 60

    n = 81562 N
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