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Physics 140 – Homework Week #13                                                                                                        Due: Wednesday, Dec

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Physics 140 – Homework Week #13                                                                                                        Due: Wednesday, Dec. 9^th   

 

10.4: Moment of Inertia Calculations

 

1(T). Four small, but dense lead balls each have a mass of 3.0 kg. They are connected into a square array by strong, but ultralight carbon fibre rods. The system is able to rotate in the plane of the paper about an axis perpendicular to the paper.

 

1. Calculate the moment of inertia of the system if the axis of rotation is at the centre of mass – located at the geometric centre of the figure (and marked with a circle)

 

2. Calculate the moment of inertia if the axis of rotation is at mass A using the basic definition of moment of inertia (i.e. NOT the parallel axis theorem)

 

3. Verify your calculation from part b) by using the result from a) and the parallel axis theorem. 

 

2.  Determine the moment of inertia of a shape made of two cylinders welded together as shown, if:

1. The cylinders are made to rotate about the centre of both cylinders
2. The cylinders are made to rotate about the edge of the large cylinder.

 

3(T). The figure to the right shows a side view of a winter tire. In this problem you’ll calculate

the moment of inertia of the rubber tire itself (e.g. not the metal wheel disk) Model it as having two sidewalls of uniform width of 0.635 cm and a tread wall of uniform thickness 2.50 cm and width 20.0 cm. Assume that the rubber has a uniform density of 1.10 x 10³ kg/m³. 

 

a)Find the moment of inertia of the car tire  about its centre. 

 

(Hint: break up the tire into three separate components and find the  volume and mass of each. Then find the moment of inertia of each  component – the tire’s overall moment of inertia will be the sum of the  individual components. Note that you are calculating the moment of inertia  for just the rubber tire and not the metal rim). 

 

b) If the tire rotates at 200 rpm, determine its rotational kinetic energy. 

 

4. A thin, 100 g disk with a diameter of 8.0 cm rotates about an axis through its centre with 0.15 J of kinetic energy. What is the speed of a point on the rim? 

 

5(T). A 25 kg solid door is 220 cm tall, 91 cm wide. What is the door’s moment of inertia for…

a) Rotation on its hinges and b) Rotation above a vertical axis inside the door, 15 cm from one edge.

c) If the door rotates at a steady angular velocity of 2 rad/s, determine its rotational kinetic energy in each of the above configurations. 

 

 

                 

10.5: Torque and Angular Acceleration

 

6. Calculate the net torque acting on the bar below due to the two forces indicated.  

               

 

7. What is the net torque about the axle on the pulley in the figure above and to the right? 

 

8. A metal rod lies  in the x-y plane with one end of the bar attached to a hinge at the origin. 

A force ??? = 7.00 ?????− 3.00 ?? ??? is applied to the bar at the point x = 3.00 m, y = 4.00 m. What is the magnitude and direction of the torque produced by this force? 

 

9. A 1.8 kg metal washer has an inner radius of 8.0 cm and and an outer radius of 12.4 cm. The washer slides over a frictionless cylindrical support so that it is free to rotate about its central axis. Determine the tangential force that must be applied to the edge of the washer in order to accelerate it from rest to 160 revolutions per minutein 20 s. 

 

10. Two 200.0 g point masses are welded onto opposite ends of a solid rod 45.0 cm long. The combined mass rotates about an axis through the rod’s centre. A torque of 0.122 N⋅m produces an angular acceleration of 4.8 rad/s². Find the mass of the bar.

 

11(T). A tradesman sharpens an axe by pushing it against the rim of a grindstone. The 30-cmdiameter stone is initially spinning at 200 rpm and has a mass of 28 kg. The coefficient of kinetic friction between the knife and stone is 0.20. If the stone loses 10 % of its speed in 10 s of grinding, what is the force with which the man presses the knife against the stone? 

12(T).  An electric motor turns a flywheel through a drive belt that joins a pulley on the motor and a pulley that is rigidly attached to the flywheel as shown in the figure below and to the left. The flywheel is a solid disk with a mass of 80.0 kg and a diameter of 1.25 m. It turns on a frictionless axle. Its pulley has a much smaller mass and a radius of 0.230 m. If the tension in the upper section of the belt is 135 N and the flywheel has a clockwise angular acceleration of 1.67 rad/s², find the tension in the lower segment of the belt. (Hint: You are only concerned with the system of disks to the left in the figure – do not concern yourself with the motor. Also note that since the pulley has a much smaller mass you can ignore the pulley for the purpose of working out the moment of inertia.)

 

                 

13. A 3 kg object on a rough (µ_s = 0.40, µ_k = 0.30) surface inclined at 25^o to the horizontal is connected to a suspended 6 kg mass via a rope and pulley. The solid steel cylindrical pulley has a radius of 10.0 cm, a width of 1.20 cm and a density of ρ = 7.80 x 10³ kg/m³. 

 

1. Determine the moment of inertia of the pulley
2. Determine the acceleration of the masses and the tensions in the ropes on either side of the pulley

 

14(T). A sphere of mass M and radius R is rigidly attached to a thin rod of radius r that passes through the sphere at a distance 1/2R from the centre. A string wrapped around the rod pulls with tension T. Find an expression for the sphere’s angular acceleration. The rod’s moment of inertia is negligible