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29 March, 18:20

Problem 4: Hydraulic systems utilize Pascal's principle by transmitting pressure from one cylinder (called the primary) to another (called the secondary). Since the pressures will be equal, if the surface areas are different then the forces applied to the cylinders' pistons will be different. Suppose in a hydraulic lift, the piston of the primary cylinder has a 1.95-cm diameter and the piston of the secondary cylinder has a 24.5-cm diameter.

What force, in newtons, must be exerted on the primary cylinder of this lift to support the weight of a 2250 kg car (a large car) resting on the secondary cylinder?

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  1. 29 March, 21:09
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    Answer: F = 139.65 N

    Explanation:

    Using Pascal principle:

    F = P * A, where

    F = Force applied to the piston

    P = Pressure

    A = Piston area

    also, we know generally,

    A = πd²/4, in this case,

    d = piston diameter

    Calculation of the force Fp necessary to support the weight of the car.

    Now, we use the Pascal principle: to solve,

    Fp = P * Ap, where

    Fp = Force on the primary piston

    Ap = Primary piston area

    we know our Ap but not P, so to find P

    We use Newton's first law for a balanced Secondary piston - automobile system

    Newton's first law: ∑F = 0

    W - Fs = 0, where

    W = car weight

    Fs = Force on the secondary piston

    Thus, W = Fs, if we put this in the equation, it becomes

    W = P * As, where

    As = Secondary piston area

    making P subject of formula,

    P = W / As

    Substituting this in the initial equation, we have

    Fp = W/As * Ap

    From this, we know that

    As = πds²/4

    Ap = πdp²/4, so that

    Fp = W / (πds²/4) * πdp²/4

    Fp = 4W/πds² * πdp²/4

    Fp = W/ds² * dp²

    Fp = 2250/24.5² * 1.95²

    Fp = 2250/600.25 * 3.8025

    Fp = 14.25 kg

    Converting to N, we have

    14.25 * 9.8 = 139.65 N
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