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EE1427 Engineering Science – Dr. Daniel Nankoo Tutorial 4 4. Rotational Kinematics and Dynamics – Tutorial 4 4.1 A spot of paint on a bicycle tire moves in a circular path of radius 0.33 m. When the spot has travelled a linear distance of 1.75 m, through what angle has the tire rotated? Give your answer in radians. (5.3 rad) 4.2 What is the angular speed (in rev/min) of the Earth as it orbits about the Sun? (1.9×10-6 rev/min) 4.3 Find the angular speed of the Earth as it spins about its axis. Give your result in rad/s. (7.27×10-5 rad/s) 4.4 One of the most studied objects in the night sky is the Crab nebula, the remains of a supernova explosion observed by the Chinese in 1054. In 1968 it was discovered that a pulsar—a rapidly rotating neutron star that emits a pulse of radio waves with each revolution—lies near the centre of the Crab nebula. The period of this pulsar is 33 ms. What is the angular speed (in rad/s) of the Crab nebula pulsar? (190 rad/s) A 3.5-inch floppy disk in a computer rotates with a period of 2.00 × 10 −1 s. What is (a) the angular speed of the disk and (b) the linear speed of a point on the rim of the disk? (c) Does a point near the centre of the disk have an angular speed that is greater than, less than, or the same as the angular speed found in part (a)? Explain. (Note: A 3.5-inch floppy disk is 3.5 inches in diameter.) (31.4 rad/s, 1.4m/s) 4.5 4.6 The angle an aeroplane propeller makes with the horizontal as a function of time is given by θ = (125 rad s)t + ( 42.5 rad s 2 )t 2 . (a) Estimate the instantaneous angular velocity at t = 0.00 s by calculating the average angular velocity from t = 0.00 s to t = 0.010 s. (b) Estimate the instantaneous angular velocity at t = 1.000 s by calculating the average angular velocity from t = 1.000 s to t = 1.010 s. (c) Estimate the instantaneous angular velocity at t = 2.000 s by calculating the average angular velocity from t = 2.000 s to t = 2.010 s. (d) Based on your results from parts (a), (b), and (c), is the angular acceleration of the propeller positive, negative, or zero? Explain. (e) Calculate the average angular acceleration from t = 0.00 s to t = 1.00 s and from t = 1.00 s to t = 2.00 s. (125 rad/s, 210 rad/s, 295 rad/s, 85 rad/s2. 85 rad/s2) 4.7 The angular speed of a propeller on a boat increases with constant acceleration from 12 rad/s to 26 rad/s in 2.5 revolutions. What is the acceleration of the propeller? (17 rad/s2) 4.8 The angular speed of a propeller on a boat increases with constant acceleration from 12 rad/s to 26 rad/s in 2.5 seconds. Through what angle did the propeller turn during this time? (48 rad) 4.9 After fixing a flat tire on a bicycle you give the wheel a spin. (a) If its initial angular speed was 6.35 rad/s and it rotated 14.2 revolutions before coming to rest, what was its average angular acceleration? (b) For what length of time did the wheel rotate? (-0.226 rad/s2, 28.1s) 4.10 A ceiling fan is rotating at 0.90 rev/s. When turned off, it slows uniformly to a stop in 2.2 min. (a) How many revolutions does the fan make in this time? (b) Using the result from part (a), find the number of revolutions the fan must make for its speed to decrease from 0.90 rev/s to 0.45 rev/s. (59, 45) 4.11 A discus thrower starts from rest and begins to rotate with a constant angular acceleration of 2.2 rad s 2 . (a) How many revolutions does it take for the discus thrower’s angular speed to reach 6.3 rad/s? (b) How much time does this take? (1.4, 2.9s) 4. Linear Rotational Kinematics and Dynamics 1 of 5 EE1427 Engineering Science – Dr. Daniel Nankoo Tutorial 4 4.12 A centrifuge is a common laboratory instrument that separates components of differing densities in solution. This is accomplished by spinning a sample around in a circle with a large angular speed. Suppose that after a centrifuge in a medical laboratory is turned off, it continues to rotate with a constant angular deceleration for 10.2 s before coming to rest. (a) If its initial angular speed was 3850 rpm, what is the magnitude of its angular deceleration? (b) How many revolutions did the centrifuge complete after being turned off? (6.29 rev/s2, 327) 4.13 A compact disk (CD) speeds up uniformly from rest to 310 rpm in 3.0 s. (a) Describe a strategy that allows you to calculate the number of revolutions the CD makes in this time. (b) Use your strategy to find the number of revolutions. (7.8) 4.14 The drill used by most dentists today is powered by a small air-turbine that can operate at angular speeds of 350,000 rpm. These drills, along with ultrasonic dental drills, are the fastest turbines in the world—far exceeding the angular speeds of jet engines. Suppose a drill starts at rest and comes up to operating speed in 2.1 s. (a) Find the angular acceleration produced by the drill, assuming it to be constant. (b) How many revolutions does the drill bit make as it comes up to speed? (2.8×103 rev/s2, 6.1×103) 4.15 The hour hand on a certain clock is 8.2 cm long. Find the tangential speed of the tip of this hand. (12µm/s) 4.16 The outer edge of a rotating Frisbee with a diameter of 29 cm has a linear speed of 3.4 m/s. What is the angular speed of the Frisbee? (23 rad/s) 4.17 A carousel at the local carnival rotates once every 45 seconds. (a) What is the linear speed of an outer horse on the carousel, which is 2.75 m from the axis of rotation? (b) What is the linear speed of an inner horse that is 1.75 m from the axis of rotation? (0.38m/s, 0.24m/s) 4.18 When a compact disk with a 12.0-cm diameter is rotating at 5.25 rad/s, what is (a) the linear speed and (b) the centripetal acceleration of a point on its outer rim? (c) Consider a point on the CD that is halfway between its centre and its outer rim. Without repeating all of the calculations required for parts (a) and (b), determine the linear speed and the centripetal acceleration of this point. (0.315m/s, 1.65m/s2, 0.158m/s, 0.827m/s2) 4.19 A Ferris wheel with a radius of 9.5 m rotates at a constant rate, completing one revolution every 32 s. Find the direction and magnitude of a passenger’s acceleration when (a) at the top and (b) at the bottom of the wheel. (0.37m/s2, 0.37m/s2) 4.20 Suppose the Ferris wheel in the previous problem begins to decelerate at the rate of 0.22 rad s 2 when the passenger is at the top of the wheel. Find the direction and magnitude of the passenger’s acceleration at that time. (2.1m/s2, 170°) 4.21 A person swings a 0.52-kg tether ball tied to a 4.5-m rope in an approximately horizontal circle. (a) If the maximum tension the rope can withstand before breaking is 11 N, what is the maximum angular speed of the ball? (b) If the rope is shortened, does the maximum angular speed found in part (a) increase, decrease, or stay the same? Explain. (2.2 rad/s) 4.22 To polish a filling, a dentist attaches a sanding disk with a radius of 3.20 mm to the drill. (a) When the drill is operated at 2.15 × 10 4 rad s , what is the tangential speed of the rim of the disk? (b) What period of rotation must the disk have if the tangential speed of its rim is to be 275 m/s? (68.8m/s, 73.1µs) 4.23 A wheel of radius R starts from rest and accelerates with a constant angular acceleration α about a fixed axis. At what time t will the centripetal and tangential accelerations of a point on the rim have the same magnitude? (√(1/α)) 4. Linear Rotational Kinematics and Dynamics 2 of 5 EE1427 Engineering Science – Dr. Daniel Nankoo Tutorial 4 4.24 To tighten a spark plug, it is recommended that a torque of 15 N ⋅ m be applied. If a mechanic tightens the spark plug with a wrench that is 25 cm long, what is the minimum force necessary to create the desired torque? (60N) 4.25 The gardening tool shown in Figure 4.25 is used to pull weeds. If a 1.23-N ⋅ m torque is required to pull a given weed, what force did the weed exert on the tool? (31N) Figure 4.25 4.26 A 1.51 kg bowling trophy is held at arm’s length, a distance of 0.605 m from the shoulder joint. What torque does the trophy exert about the shoulder if the arm is (a) horizontal, or (b) at an angle of 22.5° below the horizontal? (8.96Nm, 8.28Nm) 4.27 A person slowly lowers a 3.6-kg crab trap over the side of a dock, as shown in Figure 4.27. What torque does the trap exert about the person’s shoulder? (25Nm) Figure 4.27 4. Linear Rotational Kinematics and Dynamics 3 of 5 EE1427 Engineering Science – Dr. Daniel Nankoo Tutorial 4 4.28 A person holds a 1.42-N baseball in his hand, a distance of 34.0 cm from the elbow joint, as shown in Figure 4.28. The biceps, attached at a distance of 2.75 cm from the elbow, exerts an upward force of 12.6 N on the forearm. Consider the forearm and hand to be a uniform rod with a mass of 1.20 kg. (a) Calculate the net torque acting on the forearm and hand. Use the elbow joint as the axis of rotation. (b) If the net torque obtained in part (a) is nonzero, in which direction will the forearm and hand rotate? (c) Would the torque exerted on the forearm by the biceps increase or decrease if the biceps were attached farther from the elbow joint? (-2.14Nm) Figure 4.28 4.29 At the local playground, a 16-kg child sits on the end of a horizontal see-saw, 1.5 m from the pivot point. On the other side of the pivot an adult pushes straight down on the see-saw with a force of 95 N. In which direction does the see-saw rotate if the adult applies the force at a distance of (a) 3.0 m, (b) 2.5 m, or (c) 2.0 m from the pivot? (anticlockwise, anticlockwise, clockwise) 4.30 When a ceiling fan rotating with an angular speed of 2.65 rad/s is turned off, a frictional torque of 0.120 N ⋅ m slows it to a stop in 19.5 s. What is the moment of inertia of the fan? (0.883 kg.m2) 4.31 A person holds a ladder horizontally at its center. Treating the ladder as a uniform rod of length 3.15 m and mass 8.22 kg, find the torque the person must exert on the ladder to give it an angular acceleration of 0.302 rad s 2 . (2.05Nm) 4.32 A wheel on a game show is given an initial angular speed of 1.22 rad/s. It comes to rest after rotating through 0.75 of a turn. (a) Find the average torque exerted on the wheel given that it is a disk of radius 0.71 m and mass 6.4 kg. (b) If the mass of the wheel is doubled and its radius is halved, will the angle through which it rotates before coming to rest increase, decrease, or stay the same? Explain. (Assume that the average torque exerted on the wheel is unchanged.) (-0.25Nm) 4. Linear Rotational Kinematics and Dynamics 4 of 5 EE1427 Engineering Science – Dr. Daniel Nankoo Tutorial 4 4.33 The L-shaped object in Figure 4.33 consists of three masses connected by light rods. What torque must be applied to this object to give it an angular acceleration of 1.20 rad s 2 if it is rotated about (a) the x axis, (b) the y axis, or (c) the z axis (which is through the origin and perpendicular to the page)? (11Nm, 12Nm, 23Nm) Figure 4.33 4.34 A fish takes the bait and pulls on the line with a force of 2.1 N. The fishing reel, which rotates without friction, is a cylinder of radius 0.055 m and mass 0.84 kg. (a) What is the angular acceleration of the fishing reel? (b) How much line does the fish pull from the reel in 0.25 s? (92rad/s2, 0.16m) 4.35 Repeat the previous problem, only now assume the reel has a friction clutch that exerts a restraining torque of 0.047 N ⋅ m. (54 rad/s2, 9.3cm) 4. Linear Rotational Kinematics and Dynamics 5 of 5