Tutorial 4. Figure 1: Rod and spindle. Dog. Figure 2: I have no idea what I m doing. Dog

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Junior Elliott
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1 Tutorial 4 Question 1 Figure 1: Rod and spindle A uniform disk rotates at 3.60 rev/s around a frictionless spindle. A non-rotating rod, of the same mass as the disk and equal in length to the disk s diameter, is dropped onto the freely spinning disk as shown in Figure 1. The rod sticks to the disk and they then turn around the spindle together with their centers superposed. What is the final angular velocity (in rev/s) of the combination? [Ans:.16 rev/s] Question R Dog Figure : I have no idea what I m doing. Dog A dog of mass m is standing on the edge of a stationary horizontal turntable of rotational inertia I and radius R. The dog walks once around the circumference of the turntable, as shown in the Figure. What fraction of a full circle does the dog s motion make with respect to the ground? Express your answer in terms of m, I, and R. Ignore friction at the axle of the turntable. [Ans: I/(I + mr )] Question 3 1

2 Figure 3: (PC1431 Exam 14/15 sem 1 with major modifications + challenge problem) A basketball (which can be modelled as a hollow spherical shell) rolls down a rough mountainside into a valley and then up to the opposite side. As shown in the Figure 3, the rough part of the terrain prevents slipping while the smooth part has no friction. The initial height of the basketball is H 0. I In view of the principle of conservation of energy, explain why the ball does not return to its original height H 0 when rolling up to the other side! II How high, in terms of H 0, will the ball go up to the other side? [Ans: H = 3 5 H 0] III After reaching the maximum height in the opposite side, the ball will start to roll down and eventually enter the rough part again. Assuming that the mass of the ball is m, the coefficient of kinetic friction of the rough part is µ k, and the angle between the rough hill and the horizontal line is θ, (a) Draw the free body diagram and find the acceleration and angular acceleration when the ball enters the rough part again. [Ans: a = g[sin(θ) + µ k cos(θ)] and α = 3gµ k cos(θ) R ] (b) Consider the situation θ arctan µ k. Show that the ball will still be rotating when the ball comes to an instantaneous stop. Calculate the distance traveled when this condition is reached. [Ans: s = 3H 0 5(sin[θ)+µ k cos(θ)] ] Note: The following two parts (c and d) may be quite tedious, no need to simplify up to final expression, should at least understand the main ideas on how to do... (c) Consider now the situation θ < arctan µ k. Show that the ball will eventually reach a rolling without slipping condition while still rolling upward. Calculate the distance traveled when this condition is reached. [Ans: s = 3µ k sin(θ) 6µ k cos(θ) 6gH 0 ] g(sin(θ)+ 5 µ k cos(θ)) (sin(θ)+µ k cos(θ)) 5 (d) Hence, or otherwise, calculate the maximum height in both situations, in the case θ arctan µ k and θ < arctan µ k.

(e) Challenge Problem Assume now that tan(θ) = µ k. In general, the ball will climb up the rough hill, roll down again, reach the other side, and climb up the rough hill again.

3 (e) Challenge Problem Assume now that tan(θ) = µ k. In general, the ball will climb up the rough hill, roll down again, reach the other side, and climb up the rough hill again. This will be repeated until the ball doesn t have enough energy to climb up the rough hill. Find the distance that the ball have traveled on the rough hill in total, i.e. ignore the distance traveled on the smooth surface. [Note. There are two methods to solve this problem.] Question 4 Plank and Rollers Figure 4: Plank on two rollers A plank with a mass M = 6.00 kg rides on top of two identical solid cylindrical rollers that have radius R = 5.00 cm and mass m =.00 kg. The plank is pulled by a constant horizontal force F of magnitude 6.00 N applied to the end of the plank and perpendicular to the axes of the cylinders (which are parallel). The cylinders roll without slipping on the flat surface. There is also no slipping between the cylinders and the plank. (a) Find the acceleration of the plank and the rollers. [Ans: 0.8m/s, 0.4m/s ] (b) What static friction forces are acting? [Ans: 0.6 N, 0. N] Question 5 (For practice) Moment of inertia: Calculate the moment of inertia of a uniform solid cone as shown in Figure 5, about an axis through its center. The cone has mass M and height h. The radius of its circular base is R. [Ans: I = 3MR /10] Optional and not examinable! Only to those interested. Challenge problems are designed to be brain teasing and may involve deep understanding on the concepts with some clever tricks to solve them. If you are able to solve it, send to me your solution! The best solution will be shared to the whole class 3

4 Figure 5: A cone of height h and radius R. Question 6 (For practice) A solid cylinder of mass 1.00 kg and radius 1.41 m is rotating anti-clockwise in a uniform V-groove with constant angular acceleration α = 1.00 s, as shown in Figure 6. Figure 6: Rolling cylinder The coefficient of kinetic friction between the cylinder and each surface is What torque must be applied to the cylinder to keep it rotating at constant α? [Ans: 8.87 Nm] Question 7 (For practice) Figure 7: Collision A rigid massless rod of length L joins two particles each of mass M. The rod lies on a frictionless table, and is struck by a particle of mass M and velocity v 0, moving as 4

[Ans: ω = 4 v 0 /7L] Extra Question In this problem, you are going to consider a mathematical model of a samosa, and calculate its moment of inertia around some axes.

5 shown in Figure 7. After the collision, the projectile moves straight back. Find the angular velocity of the rod about its center of mass after the collision, assuming that mechanical energy is conserved. [Ans: ω = 4 v 0 /7L] Extra Question In this problem, you are going to consider a mathematical model of a samosa, and calculate its moment of inertia around some axes. In case you don t know what samosa is, samosa is one of my favorite snacks that tastes almost like spring rolls. It looks like this, To simplify things, let s assume that a samosa can be modeled like this instead, In the above figure, a samosa is made of two things: I The samosa dough/shell, which can be assumed to have a total mass of M, distributed uniformly in a form of a surface of an equilateral triangular prism, as shown. This problem is originally crafted by your tutor, based on one of his favorite snacks. 5

II The samosa filling, which is assumed to fully fill the above figure. The total mass of the filling is M, and it is distributed uniformly throughout the volume. For simplicity, assume l a.

6 II The samosa filling, which is assumed to fully fill the above figure. The total mass of the filling is M, and it is distributed uniformly throughout the volume. For simplicity, assume l a. Now, let s do some calculations... (a) Find the moment of inertia of an equilateral triangle with uniform linear mass density λ about an axis as shown. (b) Now, find the moment of inertia of a filled equilateral triangle with uniform area mass density σ about an axis as shown. [Hint: Parallel axis theorem can be helpful here.] (c) Using the result in part b, or otherwise, find the moment of inertia of the samosa filling about the same axis as in part b. 6

7 (d) Using the result in part a and b, or otherwise, find the moment of inertia of the samosa dough as shown previously about an axis passing through one of the corner. (e) Now, find the total moment of inertia of the samosa about the same axis as in part b and d. (f) Lastly, find the total moment of inertia of the samosa about an axis parallel to that in part e, passing through its center of mass. [Hint: Using parallel axis theorem, you don t need to repeat the whole calculation all over again.] 7

1 MR SAMPLE EXAM 3 FALL 2013

1 MR SAMPLE EXAM 3 FALL 2013 SAMPLE EXAM 3 FALL 013 1. A merry-go-round rotates from rest with an angular acceleration of 1.56 rad/s. How long does it take to rotate through the first rev? A) s B) 4 s C) 6 s D) 8 s E) 10 s. A wheel,