Chapter 10 1Page 1CHAPTER 10 - Rotational Motion About a Fixed Axis1.(a)30° = (30°)(πrad/180°) = π/6 rad = 0.524 rad;(b)57° = (57°)(πrad/180°) = 19π/60 = 0.995 rad;(c)90° = (90°)(πrad/180°) = π/2 = 1.571 rad;(d)360° = (360°)(πrad/180°) = 2π= 6.283 rad;(e)420° = (420°)(πrad/180°) = 7π/3 = 7.330 rad.2.The subtended angle in radians is the size of the object divided by the distance to the object:= 2rSun/r;(0.5°)(πrad/180°) = 2rSun/(150×106km), which gives rSun≈6.5×105km.3.We find the distance from= h/r;(7.5°)(πrad/180°) = (300 m)/r; which gives r= 2.3×103m.4.From the definition of angular acceleration, we have= ∆/∆t= [(20,000 rev/min)(2πrad/rev)/(60 s/min) – 0]/(5.0 min)(60 s/min) = 7.0 rad/s2.5.From the definition of angular velocity, we have=∆/∆t , and we use the time for each hand to turn through a complete circle, 2πrad.(a)second=∆/∆t= (2πrad)/(60 s) = 0.105 rad/s.(b)minute=∆/∆t= (2πrad)/(60 min)(60 s/min) = 1.75 ×10–3rad/s.(c)hour=∆/∆t= (2πrad)/(12 h)(60 min/h)(60 s/min) = 1.45 ×10–4rad/s.(d)For each case, the angular velocity is constant, so the angular acceleration is zero.6.(a)The Earth moves one revolution around the Sun in one year, so we haveorbit=∆/∆t= (2πrad)/(1 yr)(3.16 ×107s/yr) = 1.99 ×10–7rad/s.(b)The Earth rotates one revolution in one day, so we haverotation=∆/∆t= (2πrad)/(1 day)(24 h/day)(3600 s/h) = 7.27 ×10–5rad/s.7.All points will have the angular speed of the Earth:= ∆/∆t = (2πrad)/(1 day)(24 h/day)(3600 s/h) = 7.27 ×10–5rad/s.Their linear speed will depend on the distance from the rotation axis.(a)On the equator we havev = rEarth= (6.38×106m)(7.27 ×10–5rad/s) = 464 m/s.(b)At a latitude of 66.5° the distance is rEarthcos 66.5°, so we havev = rEarthcos 66.5° = (6.38×106m)(cos 66.5°)(7.27 ×10–5rad/s) = 185 m/s.(c)At a latitude of 40.0° the distance is rEarthcos 40.0°, so we havev = rEarthcos 40.0° = (6.38×106m)(cos 40.0°)(7.27 ×10–5rad/s) = 355 m/s.8.The subtended angle in radians is the size of the object divided by the distance to the object. A pencil with adiameter of 6 mm will block out the Moon if it is held about 60 cm from the eye. For the angle subtended we have
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