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Two wires of the same material have resistances R1 and R2. The first wire has length L1 and diameter D1. The second wire has length L2 and diameter D2. Express R2/R1 in terms of L1, L2, D1, D2.

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3 parts
15

d) 11.0Ω\text{d) } 11.0 \Omega

16

b) Zero\text{b) Zero}

17

b) Constant speed\text{b) Constant speed}

Step-by-step solution

12 steps

Part 15· 6 steps

Resistance of a wire

  1. Recall the formula for resistance.

    The resistance (RR) of a wire is given by:

    R=ρLAR = \rho \frac{L}{A}

    Where ρ\rho is the resistivity of the material, LL is the length of the wire, and AA is the cross-sectional area.

  2. Express the cross-sectional area in terms of diameter.

    The cross-sectional area of a circular wire is A=πr2A = \pi r^2, where rr is the radius. Since the diameter D=2rD = 2r, we have r=D/2r = D/2.

    A=π(D2)2=πD24A = \pi \left(\frac{D}{2}\right)^2 = \frac{\pi D^2}{4}
  3. Substitute the area into the resistance formula.

    R=ρLπD24=4ρLπD2R = \rho \frac{L}{\frac{\pi D^2}{4}} = \frac{4 \rho L}{\pi D^2}
  4. Set up a ratio for the two wires.

    Since both wires are made of the same material, their resistivity (ρ\rho) is the same. The term 4ρπ\frac{4\rho}{\pi} is constant for both wires.
    Let R1,L1,D1R_1, L_1, D_1 be the resistance, length, and diameter of the first wire, and R2,L2,D2R_2, L_2, D_2 for the second wire.

    R2R1=4ρL2πD224ρL1πD12\frac{R_2}{R_1} = \frac{\frac{4 \rho L_2}{\pi D_2^2}}{\frac{4 \rho L_1}{\pi D_1^2}} R2R1=L2D22×D12L1\frac{R_2}{R_1} = \frac{L_2}{D_2^2} \times \frac{D_1^2}{L_1} R2R1=(L2L1)×(D1D2)2\frac{R_2}{R_1} = \left(\frac{L_2}{L_1}\right) \times \left(\frac{D_1}{D_2}\right)^2
  5. Substitute the given values.

    Given:

    • L1=20.0 mL_1 = 20.0 \text{ m}
    • D1=1.50 mmD_1 = 1.50 \text{ mm}
    • R1=21.5ΩR_1 = 21.5 \Omega
    • L2=35.0 mL_2 = 35.0 \text{ m}
    • D2=3.0 mmD_2 = 3.0 \text{ mm}
    R221.5Ω=(35.0 m20.0 m)×(1.50 mm3.0 mm)2\frac{R_2}{21.5 \Omega} = \left(\frac{35.0 \text{ m}}{20.0 \text{ m}}\right) \times \left(\frac{1.50 \text{ mm}}{3.0 \text{ mm}}\right)^2 R221.5Ω=(1.75)×(0.5)2\frac{R_2}{21.5 \Omega} = (1.75) \times (0.5)^2 R221.5Ω=1.75×0.25\frac{R_2}{21.5 \Omega} = 1.75 \times 0.25 R221.5Ω=0.4375\frac{R_2}{21.5 \Omega} = 0.4375
  6. Solve for R2R_2.

    R2=0.4375×21.5ΩR_2 = 0.4375 \times 21.5 \Omega R2=9.40625ΩR_2 = 9.40625 \Omega

    Comparing this value to the options, 9.40625Ω9.40625 \Omega is closest to 11.0Ω11.0 \Omega.

    The resistance of the second wire is approximately d) 11.0Ω\boxed{\text{d) } 11.0 \Omega}.


Part 16

b) Zero\text{b) Zero}

Acceleration of an object with constant velocity

  1. Understand the definition of acceleration.

    Acceleration is the rate of change of velocity.

    a=ΔvΔta = \frac{\Delta v}{\Delta t}
  2. Apply the definition to constant velocity.

    If an object moves with a constant velocity, it means its velocity is not changing. Therefore, the change in velocity (Δv\Delta v) is zero.

  3. Calculate the acceleration.

    Since Δv=0\Delta v = 0, the acceleration (aa) must be zero.

    a=0Δt=0 m/s2a = \frac{0}{\Delta t} = 0 \text{ m/s}^2

    The acceleration of the object is b) Zero\boxed{\text{b) Zero}}.


Part 17

b) Constant speed\text{b) Constant speed}

Distance directly proportional to time

  1. Interpret "distance covered by an object is directly proportional to time".

    This means that if dd is distance and tt is time, then:

    d∝td \propto t

    This can be written as d=ktd = k t, where kk is a constant of proportionality.

  2. Relate this to velocity.

    Velocity (vv) is defined as the rate of change of distance with respect to time:

    v=dddtv = \frac{dd}{dt}

    If d=ktd = k t, then differentiating with respect to time gives:

    v=ddt(kt)=kv = \frac{d}{dt}(kt) = k

    Since kk is a constant, the velocity (vv) is constant.

  3. Conclude the type of motion.

    An object moving with constant velocity is said to travel with constant speed (assuming the direction is also constant, which is implied by "distance covered" in a single direction).

    The object is said to travel with b) Constant speed\boxed{\text{b) Constant speed}}.

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