Physics Exercises: Volleyball Serve & Disk Mechanics Problems

PHYSICS
Exercise 1
During a volleyball match, a student recorded a video sequence of the motion of the ball from the
moment the serve was
executed at point ,
located at a height above
the ground. The player
who performed the serve is
positioned at a horizontal
distance  from the net
(Figure 1).
For the serve to be valid,
the ball must satisfy the
following two conditions:
-Pass over the net, whose top is at a height  above the ground;
-Land in the opponent’s court, which has a length 
Data : Air resistance and the size of the ball are negligible. The ball can be considered a point
mass. Gravitational acceleration:    m.s .    m,      m,    m.
We study the motion of the ball in an orthonormal reference frame   attached to the Earth
and assumed to be Galilean.
At time   , the ball is launched from point with an initial velocity  making an angle with
the horizontal (Figure 1).
Using appropriate software, the video was analyzed to obtain the curves shown in Figure 2. The
graphs and represent the time variation of the components of the velocity vector of the
ball in the reference frame   .
1. By applying Newton’s second law, find the
expressions of the velocity components and
as functions of time.
2. Using the curves in Figure 2, determine that:
 m.sand   
3.Determine the equation of the trajectory  in
the reference frame  
4. Assuming the ball is not intercepted by any
player, determine whether it satisfies the two
conditions required for a valid serve. Justify your
answer.

Exercise 2
Disks are essential components in mechanical systems used in agriculture and industry. They are
characterized by their moment of inertia.
This exercise aims to:
-determine the moment of a friction torque;
-study the effect of variation of a disk’s radius on angular
acceleration.
We consider a homogeneous solid disk in a mechanical
device. This disk has mass  and radius , and it can rotate
about a fixed horizontal axis passing through its center
(Figure 1).
The moment of inertia of the disk about is: 
At time   , the disk , initially at rest , is set into motion by a motor that applies a constant
driving torque . The friction between the axis and the disk is modeled by a constant
resisting torque .
Data:    
1. By applying the fundamental law of dynamics for rotation about a fixed axis, show that the
angular acceleration of the disk is given by:
2. An experimental study provides the graph of the
angular velocity  of the disk (Figure 2).
Graphically determine the value of the angular
acceleration .
3. Write the time equation of the motion of the
disk.
4. Determine the value of the friction torque .
5. Determine the tangential acceleration and normal acceleration of a point located at
the rim of the disk at time    s.
6. The motor is switched off at    min. At that moment, the angular speed of the disk
is:  The disk then stops after a time interval .
Determine , the braking time.
7. We consider a second disk of the same material, with mass  , radius 
, and
moment of inertia about : 
.It is assumed that is subjected to the same
driving torque and the same resisting torque .The angular acceleration of disk is
denoted .Compare and . What can we conclude?