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Tutorial Problem Set 1 16361/ME305 Dynamics 3

1. The position of a particle s during an interval of time t = [0, 12] sec is given by 1 s = t 3 + 6t 2 + 4t 2 and is measured in metres. a) What distance has the particle travelled after 6 seconds? After 8 seconds? b) What is the maximum velocity during the time interval t = [0, 12] sec, and at what time does it occur? c) What is the acceleration of the particle at the moment when the velocity is at the maximum? 2. A rocket starts from rest and travels straight up. Its height is measured by a radar from t = 0 sec to t = 4 sec and is estimated to be s = 10t2 . a) What height has the rocket reached after 3 sec? b) What is the velocity at t = 4 sec? c) What is the acceleration during the first four seconds? 3. A train accelerates uniformly from rest to reach 54 km/h in 200 sec after which the speed remains constant for another 300 sec. At the end of this time the train decelerates to rest in 150 sec. What is the total distance travelled by the train? 4. The cheetah can travel as fast as 105 km/h. If you assume that the animal’s acceleration is constant and it reaches top speed in 4 sec, what distance it can cover in 10 sec? 5. A vehicle was lifted off the ground and suspended in air before being dropped to the ground. The vertical velocity of a vehicle when it reached the ground was v = 20 m/s. From what height was the vehicle dropped from? How long did it take to reach the ground?

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Tutorial Problem Set 1 with Full Solutions 16361/ME305 Dynamics 3

1. The position of a particle s during an interval of time t = [0, 12] sec is given by 1 s = t 3 + 6t 2 + 4t 2 and is measured in metres. (a) What distance has the particle travelled after 6 seconds? After 8 seconds? (b) What is the maximum velocity during the time interval t = [0, 12] sec, and at what time does it occur? (c) What is the acceleration of the particle at the moment when the velocity is at the maximum? 2. A rocket starts from rest and travels straight up. Its height is measured by a radar from t = 0 sec to t = 4 sec and is estimated to be s = 10t2 . (a) What height has the rocket reached after 3 sec? (b) What is the velocity at t = 4 sec? (c) What is the acceleration during the first four seconds? 3. A train accelerates uniformly from rest to reach 54 km/h in 200 sec after which the speed remains constant for another 300 sec. At the end of this time the train decelerates to rest in 150 sec. What is the total distance travelled by the train? 4. The cheetah can travel as fast as 105 km/h. If you assume that the animal’s acceleration is constant and it reaches top speed in 4 sec, what distance it can cover in 10 sec? 5. A vehicle was lifted off the ground and suspended in air before being dropped to the ground. The vertical velocity of a vehicle when it reached the ground was v = 20 m/s. From what height was the vehicle dropped from? How long did it take to reach the ground?

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16361/ME305 Tutorial Set 1 with Full Solutions

Solutions 1. The position of a particle s during an interval of time t = [0, 12] sec is given by 1 s = t 3 + 6t 2 + 4t 2 and is measured in metres. (a) What distance has the particle travelled after 6 seconds? after 8 seconds? s(t = 6) = 12 (6)3 + 6(6)2 + 4(6) = 348 m s(t = 8) = 672 m (b) What is the maximum velocity during the time interval t = [0, 12] sec, and at what time does it occur? v=

ds 3 2 = t + 12t + 4 dt 2

a=

d2 s dv = 2 = 3t + 12 dt dt

The maximum (or minimum) velocity will occur at when the acceleration is equal to 0. Imagine if you throw a ball upwards, the maximum height occurs at the top of the parabola when the velocity is zero. This is the same principle with velocity and acceleration. Therefore we can find the time at which the velocity is at maximum by setting a = dv/dt = 0 and solving for t, a = 3t + 12 = 0 → t = −4 sec, which not a valid answer and moreover is outside of the range of time given. Another way is to examine the equation for the acceleration; it can be easily seen that it is linearly increasing, therefore the maximum velocity occurs at the maximum time, which here is t = 12 sec. This can be seen in Fig. 1, which shows the plots of the position, velocity and acceleration relative to time. (c) What is the acceleration of the particle at the moment when the velocity is at the maximum? Since the maximum velocity occurs at t = 12 sec, then the acceleration at that time is a(t = 12) = 3(12) + 12 = 48 m/s2 . 2. A rocket starts from rest and travels straight up. Its height is measured by a radar from t = 0 sec to t = 4 sec and is estimated to be s = 10t2 . (a) What height has the rocket reached after 3 sec? s(t = 3) = 90 m (b) What is the velocity at t = 4 sec? v=

ds = 20t −→ v(t = 4) = 20(4) = 80 m/s dt 2

16361/ME305 Tutorial Set 1 with Full Solutions

(c) What is the acceleration during the first four seconds? a=

dv 2 = 20 m/s , which is independent of time. dt

3. A train accelerates uniformly from rest to reach 54 km/h in 200 sec after which the speed remains constant for another 300 sec. At the end of this time the train decelerates to rest in 150 sec. What is the total distance travelled by the train?

The velocity profile can be divided into three sections (A, B and C) as shown above. The velocity vmax = 54 km/hr has to converted into consistent units, vmax =

1 hr 54 km 1000 m · · = 15 m/s 1 km 1 hr 3600 sec

For the first section A from t = [0, 200] sec: ∆v vmax − 0 2 = 0.0750 m/s = 200 ∆t 1 1 ∆sA = s0 + v0 ∆t + a∆t2 = aA (200)2 = 1500 m 2 2 aA =

For the second section B with a constant velocity vmax from t = [200, 500] sec: ∆sB = vmax∆t2B = 15(300) = 4500 m For the third section C from t = [500, 600] sec the train is decelerating therefore the acceleration should be negative, however in this case we are just concerned with determining the magnitude of the distance travelled: 0 − vmax ∆v 2 = −0.100 m/s = 650 − 500 ∆t 1 1 ∆sA = |a|∆t2 = (0.100)(150)2 = 1125 m 2 2 aC =

The sum of the three distances gives the total distance travelled:

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16361/ME305 Tutorial Set 1 with Full Solutions

X

s = sA + sB + sC = 7125 m

4. The cheetah can travel as fast as 105 km/h. If you assume that the animal’s acceleration is constant and it reaches top speed in 4 sec, what distance it can cover in 10 sec?

The acceleration during the first 4 sec that it takes to reach the maximum velocity is given by, 29.167 ∆v dv 2 = 7.292 m/s = = aA = 4 ∆t dt where vmax = 105 km/h = 29.167 m/s. Therefore the distance travelled during the initial time from t = [0, 4] sec is, 1 sA = aA ∆t = 58.333 m 2 After the cheetah reaches the maximum velocity, it continues at that same constant speed for the next 6 sec (or until t = 10 sec). sB = ∆vmax∆t = 29.167(6) = 175 m Therefore the total distance covered is, X s = sA + sB = 233.33 m 5. A vehicle was lifted up and suspended in air before being dropped to the ground. The vertical velocity of a vehicle when it reached the ground was v = 20 m/s. From what height was the vehicle dropped? How long did it take to reach the ground? vf2 = vi2 + 2a∆s −→ ∆s =

vf2 − v2i 2a

If we neglect any drag on the vehicle and assume that the only acceleration acting on the

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16361/ME305 Tutorial Set 1 with Full Solutions

object is gravity, then vehicle was dropped from a height of, s=

−(20)2 = 20.387 m 2(−9.81)

and took t = ∆s/∆v = 1.02 seconds to reach the ground.

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16361/ME305 Dynamics 3

Tutorial Problem Set 2 1. The crank OA of the engine mechanism is rotating counterclockwise at 3600 rpm. (a) What is the velocity of A and its acceleration if OA = 100 mm? (b) What is the angular and the linear distance traveled by A over a period of 10 sec?

2. The armature of an electric motor rotates at a constant rate. The magnitude of the velocity of point P relative to O is 4 m/s, and R = OP = 80 mm. (a) What are the normal and the tangential acceleration of P relative to O ? (b) What is the angular velocity of the armature?

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Tutorial Problem Set 2

3. Suppose you want to design a centrifuge to subject samples to centripetal acceleration of 1000 g. (a) What speed of rotation is necessary if the distance of the sample from the center is 300 mm? (b) If you want the centrifuge to reach its design speed in 1 min, what constant angular acceleration is needed? 4. The angular displacement θ that OP covers is given by θ = 2t2 rad, where OP = 1.2 m. (a) What are the magnitude of the linear velocity and the acceleration of P relative to O at t = 1 sec? (b) What distance along the circular path does P move from t = 0 to t = 1 sec?

5. The bent rod rotates around a line joining Points A and E, with a constant angular velocity of 9 rad/s. Knowing the rotation is clockwise (as viewed from Point E), determine the velocity and acceleration of Point C. (Problem 15.10 in Vector Mechanics for Engineers, 10th Edition)

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Tutorial Problem Set 2

6. At the instant shown, the angular velocity of rod AB is 15 rad/s clockwise. Determine (a) the angular velocity of rod BD, and (b) the velocity of the midpoint of rod BD. (Problem 15.63 in Vector Mechanics for Engineers, 10th Edition )

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16361/ME305 Dynamics 3

Tutorial Problem Set 2 Questions 1. The crank OA of the engine mechanism is rotating counterclockwise at 3600 rpm. (a) What is the velocity of A and its acceleration if OA = 100 mm? (b) What is the angular and the linear distance traveled by A over a period of 10 sec?

2. The armature of an electric motor rotates at a constant rate. The magnitude of the velocity of point P relative to O is 4 m/s, and R = OP = 80 mm. (a) What are the normal and the tangential acceleration of P relative to O ? (b) What is the angular velocity of the armature?

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Tutorial Problem Set 2

3. Suppose you want to design a centrifuge to subject samples to centripetal acceleration of 1000 g. (a) What speed of rotation is necessary if the distance of the sample from the center is 300 mm? (b) If you want the centrifuge to reach its design speed in 1 min, what constant angular acceleration is needed? 4. The angular displacement θ that OP covers is given by θ = 2t2 rad, where OP = 1.2 m. (a) What are the magnitude of the linear velocity and the acceleration of P relative to O at t = 1 sec? (b) What distance along the circular path does P move from t = 0 to t = 1 sec?

5. The bent rod rotates around a line joining Points A and E, with a constant angular velocity of 9 rad/s. Knowing the rotation is clockwise (as viewed from Point E), determine the velocity and acceleration of Point C. (Problem 15.10 in Vector Mechanics for Engineers, 10th Edition)

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Tutorial Problem Set 2

6. At the instant shown, the angular velocity of rod AB is 15 rad/s clockwise. Determine (a) the angular velocity of rod BD, and (b) the velocity of the midpoint of rod BD. (Problem 15.63 in Vector Mechanics for Engineers, 10th Edition )

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Tutorial Problem Set 2

Solutions 1. The crank OA of the engine mechanism is rotating counterclockwise at 3600 rpm.

(a) What is the velocity of A and its acceleration if OA = 100 mm? ωA =

(3600 rpm)(2π rad/rev) = 376.8 rad/s 60 sec/min

∴ vA = ωA r = 37.68 m/s where r = OA = 0.1 m. aA = atˆt + ann ˆ dv =0 (tangential component of the acceleration) dt 2 v an = A = 14197.82 m/s2 (normal component of the acceleration) r at =

(b) What is the angular and the linear distance travelled by A over a period of 10 sec? The angular distance travelled is θA = ωA ∆t = 1200π rad. The linear distance is sA = vA ∆t = 376.8 m. 2. The armature of an electric motor rotates at a constant rate. The magnitude of the velocity of point P relative to O is 4 m/s, and R = OP = 80 mm.

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Tutorial Problem Set 2

(a) What are the normal and the tangential acceleration of P relative to O ? dv =0 dt v2 (aP )n = P = 200 m/s2 r

Tangential acceleration:

(aP )t =

Normal acceleration: (b) What is the angular velocity of the armature? ωP =

4 vP = = 50 rad/s = 7.96 rev/s r 0.08

3. Suppose you want to design a centrifuge to subject samples to centripetal acceleration of 1000 g. (a) What speed of rotation is necessary if the distance of the sample from the center is 300 mm? Given the normal acceleration an : 2

2

an = 1000 g = 1000(9.807 m/s ) = 9807 m/s v2 √ −→ v = an r = 54.22 m/s r v ∴ ω = = 180.74 rad/s r =

(b) If you want the centrifuge to reach its design speed in 1 min, what constant angular acceleration is needed? Since the acceleration is constant, then α = ω/∆t = 180.74/60 = 3.01 rad/s2 . 4. The angular displacement θ that OP covers is given by θ = 2t2 rad, where OP = 1.2 m.

(a) What are the magnitudes of the linear velocity and acceleration of P relative to O at t = 1 sec? We start with the equation for the angular displacement and differentiate to get the

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Tutorial Problem Set 2

equations for the angular velocity and acceleration, d dθ = (2t2 ) = 4t dt dt dω d 2 α= (4t) = 4 rad/s = dt dt

ω=

Given the relation v = ωr, therefore the linear velocity is vP (t = 1) = (4t)r = 4(1)(1.2) = 4.8 m/s. The two components for the acceleration are, dv d 2 = (4tr) = 4t −→ at (t = 1) = 4 m/s dt dt v2 2 an = = 19.2 m/s r q at =

∴ kaP k = aP =

a2n + at2 = 19.79 m/s2

(b) What distance along the circular path does P move from t = 0 to t = 1 sec? s = θr = 2t2 r −→ sP (t = 1) = 2(1)2 (1.2) = 2.4 m 5. The bent rod rotates around a line joining Points A and E, with a constant angular velocity of 9 rad/s. Knowing the rotation is clockwise (as viewed from Point E), determine the velocity and acceleration of Point C.

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Tutorial Problem Set 2

Looking first at determining the position vectors from points C to E, and E to A. rC/E = −0.4 ˆi + 0.15 ˆ j

j + 0.2 ˆk rE/A = −0.4 ˆi + 0.4 ˆ p krEA k = rEA = 0.42 + 0.42 + 0.22 = 0.6 m To calculate the vector of the rotational velocity ω, which follows the right hand rule, requires using the unit vector for the position (ˆ r = r/r). From there, you can find the translational velocity and acceleration at Point C. k rEA −0.4ˆi + 0.4ˆj + 0.2ˆ = (−6ˆi + 6ˆ j + 3ˆ k) rad/s =9 rEA 0.6 ˆ m/s ∴ vC = ω AE × rC/E = −0.45ˆi − 1.2ˆj + (−0.9 + 2.4)ˆk = −0.45ˆi − 1.2ˆj + 1.5k ω EA = ωEA

∴ aC = αAC × rC/E + ω AE × (ω AE × rCE ) = αAC × rC/E + ω AE × vC

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ˆ m/s i + (−1.35 + 9)ˆj + (7.2 + 2.7)ˆk = 12.6ˆi + 7.65ˆj + 9.9k = (9 + 3.6)ˆ

6. At the instant shown, the angular velocity of rod AB is 15 rad/s clockwise. Determine (a) the angular velocity of rod BD, (b) the velocity of the midpoint of rod BD.

Starting with rod AB, points A and B have the same angular velocity ωA = ωB = 15 rad/s, ˆ rad/s. The position vector of point B relative to point A is rB/A = −0.2ˆj or ω AB = −15k m, and vA = 0 since it is fixed. The velocity of point B is then: vB = vA + vB/A = 0 + (ω BA × rB/A ) = −3ˆi m/s

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Tutorial Problem Set 2

Similarly, for rod BD, rB/D = 0.6ˆi + 0.25ˆj vB = vD + vB/D = vD + (ω BD × rB/D )    −3ˆi = vDˆj + ωBD ˆk × 0.6ˆi + 0.25ˆj = −0.25ωBDˆi + (vD + 0.6ωBD )ˆj ˆ gives Equating the two sides for the coefficients for ˆi and j, ω BD = 12 ˆk rad/s

vD = −7.2ˆj m/s

Taking the midpoint C of rod BD, rC/D = 0.5rB/D = 0.3ˆi + 0.125ˆj    vC = vD + vC/D = −7.2ˆj + 12 ˆk × 0.3ˆi + 0.125ˆj

∴ vC = (−1.5ˆi − 3.6ˆj) m/s, and vC = 3.9 m/s

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16361/ME305 Dynamics 3

Tutorial Problem Set 3 1. A planning machine table of mass 450 kg attains a speed of 0.6 m/s in a distance of 500 mm from rest, with uniform acceleration. The coefficient of friction between table and bed is 0.1. Calculate the required force applied to the table. 2. A mass of 1 kg is hung from a spring balance in a lift. What is the spring balance reading (i.e. net force) if: (a) the lift is at rest? (b) the lift is accelerating upwards at 3 m/s2 ? (c) the lift is accelerating downwards at 3 m/s2 ? (d) moving downwards but decelerating at 3 m/s2 ? 3. A car of mass 1.1 (metric) tonnes is driven at constant speed up an incline of 10◦ . The rolling resistance is 170 N and the air resistance is 100 N. (a) Find the tractive effort (i.e., the driving force of the engine)? (b) What is the effort when the car has a uniform acceleration 0.6 m/s2 ? 4. Coal wagons are lowered 30 m from rest down a slope of 10◦ with uniform acceleration, by means of a cable attached to a rope brake. The total mass of the wagons is 48 tonnes and the resistance to motion is 130 N/ton. At the end of the incline the wagons are travelling at 3 m/s. What is the pull in the cable? 5. A cast iron pulley is 200 mm wide and 25 mm thick with a mean diameter of 2 m. Consider the pulley as a thin ring and find its moment of inertia. What torque is required to produce a pulley speed of 5 rev/s in 15 seconds. The density of cast iron is ρ = 7200 kg/m3 . 6. A short heavy shaft is being turned in a lathe which is driven by a motor running at 1400 rpm, and the speed reduction between the motor and the lathe spindle is 10:1. The motor torque is constant at 15.35 Nm, and the friction torque at the lathe spindle is constant at 17.5 Nm. The mass moment of inertia of the rotating parts within the motor is 0.08 kg·m2 , and that of the lathe face-plate, spindle and work piece being turned in a total of 1.2 kg·m2 . Assume that the turning tool is given an excessively heavy cut that stops the work piece in one revolution, and then proceed to calculate the total force on the tip of the tool if it is cutting at a radius of 140 mm. You can assume the tool tip force to be uniform during the deceleration period.

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Tutorial Problem Set 3

7. Two shafts, A and B, lie parallel, their axes being 50 cm apart. Each carries a pulley of diameter 30 cm and the pulleys are connected by a belt so that power can be transmitted from one shaft to the other. If each pulley and shaft has a combined mass moment of inertia about its axis of 0.05 kg·m2 and if the mass per unit length of the belt is 0.4 kg/m, determine the kinetic energy when the angular velocity of the shaft is 100 rpm. If the system is initially at rest, what constant torque must be applied about A to accelerate the system to the above angular velocity after 10 revolutions? 8. A single aircraft wheel is of mass 125 kg, diameter 0.9 m, and radius of gyration 0.375 m. As it approaches the runway the wheel is completely without any angular velocity, but once contact is made the friction between the tyre and the runway causes it to rotate. Exactly one second after the initial contact the wheel rolls without slipping at an angular velocity which conforms to the landing speed of the aircraft. If the forward speed of the landing aircraft is 50 m/s at the instant when slipping between the tyre and the runway ceases, determine the following for the single wheel: (a) its kinetic energy, (b) the angular momentum about its axis of rotation, (c) the magnitude of the frictional force between the runway and the tyre, assuming this to be constant during the interval between the initial touchdown and the cessation of slipping. 9. In the system below, Disc 1 has a mass of m1 , a radius r1 , while Disc 2 has a mass of m2 , a radius of r2 . Both discs have moment of inertia about its central axis of I = (1/2)mr2 . Both discs can rotate and slide up and down in the frictionless slot in the fixed frame as shown. Disc 1, having an angular velocity of ω rad/s clockwise, is dropped from a negligible hei...