Tute M4 : ROTATIONAL MOTION 1 The equations dealing with rotational motion are identical to those of linear motion in their mathematical form. To convert equations for linear motion to those for rotational motion simply replace the linear quantities with their corresponding rotational analogues. A brief summary is presented below. Linear and rotational quantities: Displacement x Velocity v Acceleration a Mass m Force F v = dx v = v 0 + at v 2 = v 2 0 + 2ax x = v 0 t + 1 2 at2 a = dv = v dv = d2 x dx 2 p = mv F = dp K = 1 2 mv2 F = ma W = F x P = F v F = kx U = 1 2 kx2 T = 2π m k Angular displacement θ Angular velocity ω Angular acceleration α Moment of inertia I Torque τ ω = dθ ω f = ω i + αt ω 2 f = ω2 i + 2αθ θ = ω i t + 1 2 αt2 α = dω = ω dω = d2 θ dθ 2 L = Iω τ = dl K = 1 2 Iω2 τ = Iα W = τθ P = τω τ = κθ U = 1 2 κθ2 T = 2π I κ 1 September 20, 2017 1
1. The flywheel of a car idles at 600 revolutions per minute (rpm). Pressing the accelerator pedal results in it spinning up to 2400 rpm uniformly in 1.6 s. Calculate the angular acceleration of the flywheel. [ Answer: α = 117.8 rads 2 ] 2. Calculate the centripetal acceleration for the (a) Moon orbiting the Earth (b) Earth orbiting the Sun. Data: Moon-Earth distance = 3.84 10 8 m Earth-Sun distance = 1.4957 10 11 m Orbital period of the Moon = 27.32 days. [ Answer: (i) a ME = 2.72 mm s 2 (ii) a ES = 5.93 mm s 2 ] 3. Calculate the speed and acceleration of a person in Quito (on the equator) resulting from the earth s spinning about its axis through the poles. The radius of the equator is 6.37 10 6 m. What if the person is located on the Hovgaard Island a latitude 80 north of the equator? [ Answer: (i) 463 m/s, 0.0337 m/s 2 ; 80.44ṁ/s, 0.006 m/s 2 ] 4. After a drill is switched off the tip decelerates uniformly from 5000 rpm to a complete stop in 5 s. Find the angular deceleration of the drill bit and the angle turned through in the 5 second period. [ Answer: α = 104.7 rad s 2 ; θ = 1309.2 radians = 208.4 revolutions ] 5. A bus, of mass 200 kg, is going around a 10 m radius round-about. The bus, initially at rest, starts from the north and goes clock-wise around the round-about. It will take the bus 3 s to reach a speed of π rad/s. Find (a) the angular and tangential acceleration of the bus. 2
(b) the angular displacement, displacement vector and distance travelled by the bus in that time interval. (c) the centripetal force acting on the bus at t = 3 s. (d) velocity of the bus relative to it s velocity at t = 2.45 s. [ Answer: α = π 3 rad/s2, 10π 3 m/s2 3π ; 2-8.2πi + 10πj m/s ] rad, 10i - 10j m, 30π 3 m ; 2000π2 N ; 6. The position, velocity and acceleration vectors for a particle undergoing circular motion are shown. Is the particle travelling clock-wise or anti-clock-wise in each case? Is its speed increasing or decreasing? Figure 1 [ Answer: ccw, decreasing speed ; cw, constant speed ] 7. An object starts from rest, and takes 2 s to move counter-clock-wise from the x-axis to θ above the axis. Its speed at 2 s is 0.7 m/s. Find (a) its angular velocity and angular acceleration at 2 s. (b) its tangential acceleration and centripetal at 2 s. (c) the distance travelled after 2 s. (d) its position vector at 2 s. (e) the displacement vector after 2 s. Figure 2 [ Answer: 0.175 rad/s, 0.088 rad/s 2 ; 0.35 m/s 2, 0.123 m/s 2 ; 0.7 m; 3.94i + 0.696j m; -0.06i + 0.696j m ] 3
ω A A B t = 0 Figure 1 6. An arrow is aimed at a dart-board 2 m away, level with the x-axis. The dart-board is rotating anti-clockwise at a constant angular velocity of 5 rad/s. The radius of the dart-board r = 0.4 m and point A lies just at the rim of the board. The arrow is released from its bow with a horizontal speed of 8 m/s the moment point A is lined up along the x-axis as shown. The arrow hits the dart-board at point B, which lies along the x-axis but below A. Neglecting any gravitational or frictional forces on the arrow, find the vertical height AB. [ Answer: 0.23 m ] 7. If in the above question, the gravitational force is NOT negligible, find the angular velocity of the dart-board if the arrow now hits a mark 0.5 m directly below the point A. [ Answer: 3.96 rad/s ] 8. Two pebbles are placed on a turntable which is spinning clockwise at a constant angular velocity ω. Pebble A is placed at a distance r from the center while pebble B is placed at a distance of 2r from the center. What is the ratio of the linear velocity for pebble A to that for pebble B? [ Answer: 1 : 2 ] 9. A ladybug is walking around the rim of a plate of radius r = 20 cm in an anticlockwise direction at a constant speed of v = 0.25 m/s. If the ladybug starts from point P 3
on the rim, express in i, j, k, unit vectors, the position vector of the ladybug at anytime t relative to point P. What is the angular velocity ω of ladybug around the rim? How long will it take it to complete one circle? [ Answer: r = 0.2 cos(1.25t)i + sin(1.25t)j), ω = 1.25 rad/s, t = 1.6π s ] 10. A stone, attached to a wheel and held in place by a string, is whirled in circular orbit of radius R in a vertical plane. Suppose the string is cut when the stone is at position 2 in Figure 2, and the stone then rises to a height h above the point at position 2. What was the angular velocity of the stone when the string was cut? Give your answer in terms of R, h and g. [ Answer: ω = 2ghR 2 ] Figure 2 11. A bead moves along a circular wire. Its speed increases at a = 2 t 4 m/s 2. Its initial (at t = 0) position and speed are s 0 = 0 m and v 0 = 3 m/s. At t = 5 s, determine: (a) The magnitude of the beads acceleration. (b) The position of the bead along the wire (give both arclength s, and angle, θ). (c) The total distance traveled along the wire by the bead in the 5 s time interval. Figure 3 [ Answer: a = a 2 t + a 2 r = 6.8 m/s 2, s = 6.67 m θ = 19.1 ] 4