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Tutorial 6
Chater 6: Mechamics of Rigid Bodies
Exercice 1:
1. Obtain the moment of inertia tensor of a thin uniform ring of radius R, and mass
M, with the origin of the coordinate system placed at the center of the ring, and the
ring lying in the xy plane.
2. Obtain the moment of inertia tensor of a uniform hollow sphere of radius R, and
mass M, with the origin of the coordinate system placed at the center of the sphere.
Note that this problem can be done by dividing the sphere into a large number of
infinitesimally thin rings.
Exercice 2:
1. Find the moments of inertia of a disk of radius R, thickness t, total mass M, and
total volume V shown in the image below.
Figure 1: Disk rotating about central axis.
2. Find the moment of inertia of a disk rotating about an axis passing through the
disk’s circumference and parallel to its central axis, as shown below. The radius of
the disk is R, and the mass of the disk is M.
3. Obtain the moment of inertia tensor of a uniform solid sphere of radius R, and
mass M, with the origin of the coordinate system placed at the center of the sphere.
Dr DJOB
Physics for engineering I
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Figure 2: Disk rotating about an axis passing through the circumference.
Note that this problem can be done by dividing the sphere into a large number of
infinitesimally thin disks.
Exercice 3:
A solid wheel has a mass of 5 kg and outer radius of 200 mm. Calculate the torque
requiered to accelerate it from rest to 1500 rev/min in 10 seconds.
Exercice 4:
The flywheel of a stationary exercise bicycle is made of a solid iron disk of radius 0.2
m and thickness 0.02 m. A person applies a torque that has an initial value of 25 Nm and
decreases at the rate of 5 Nm/s for a total time of 5 seconds.
1. What is the moment of inertia of the wheel?
2. What is the initial angular acceleration of the wheel?
3. What is the rate at which the person needs to pedal after 3 seconds to be able to
continue applying the torque?
4. How much energy has the person converted to the rotational kinetic energy of the
wheel after 5 seconds?
Exercice 5:
1. Find the inertia tensor of a homogeneous cube of side b relative to its corner.
2. Calculate the components Iij of the inertia tensor relative to the centre of mass.
Dr DJOB
Physics for engineering I
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Exercice 6:
Find the kinetic energy of a wheel rotating at 4 rad/s given the mass is 3 kg and the
radius of gyration 0.2 m.
Exercice 7:
A body S of mass 80kg is released on a plane inclined at an angle of 30 with respect
to the horizontal and it slides downwards without friction
1. Apply the second Newton’s law to find the coordinates of the acceleration vector in
the frame (O, x, y) where x-axis is parallel to the plane and in the direction of the
motion and the y-axis perpendicular to the x-axis
2. Knowing that the body S leaves at time t=0 from the anointed A with a speed
vA =5m/s (A is coincident with the origin O of the space reference
(a) Write down the equations of motion of S along the axis x-axis during time then
the equation of its velocity
(b) Determine its velocity at B point located at 2 meters
g=10m/s2
Dr DJOB
Physics for engineering I