Technical mechanics I
Type 1
1, The small roller on rope AB is in static equilibrium. The m mass of the load which is hanging on the
roller and all the necessary geometric data are given.
Data: mg = 800 N; a = 6 m; b = 2 m; c = 1 m.
a
b
A
B
c
F
mg
a, Calculate the magnitude of force F.
b, Check your result with construction.
2, A particle with mass m is connected to a spring and can slip on the smooth surfaced rod without
friction. The D constant of the spring is given, and its extension is zero when its direction is normal to
the rod.
Data: h=0,2[m] , α=15°, s=0,05[m], D=15000[N/m], m= 7[kg].
D
F
m
h
P
s
α
a, Calculate the extension of the spring when the particle is in point P.
b, Calculate the magnitude of force F when the particle is in static equilibrium in point P.
c, Construct the magnitude of force F
3, A particle with mass m is connected to a spring and can slip on the smooth surfaced incline
without friction. The D constant and r compression of the spring is given below.
Data:  = 30°, r = 2 mm, D = 103 N/mm, m = 40 kg.

D
F
m

a, Calculate the magnitude of the horizontal force F if the particle is in static equilibrium.
b, Calculate the magnitude of the reaction force F n which the incline exerts on the particle.
c, Construct F and F n .
Type 2
1,
a, Calculate the coordinates of the centre of gravity of the plate in the figure.
b, Check your results with construction.
Data: a = 30 mm; b = 50 mm; c = 20 mm.
y
a
c
O
x
c
b
2,
a, Calculate the coordinates of the centre of gravity of the plate in the figure.
b, Check your results with construction.
Data: a = 30 mm; b = 50 mm; R = 15 mm.
y
R
a
O
b
x
3,
a, Calculate the coordinates of the centre of gravity of the plate in the figure.
b, Check your results with construction.
Data: a = 5 cm; b = 3 cm; c = 1 cm; d = 1 cm.
c
y
c
Ød
a
c
O
x
b
Type 3
1,
Calculate and construct the F A and F B reaction forces that act on the rigid plate in the figure.
Data: F1 = 10 kN; F2 = 5 kN; a = 400 mm; b = 200 mm; c = 500 mm; d = 200 mm.
F1
A
b
F2
a
B
d
c
2,
Calculate and construct the F A and F B reaction forces that act on the rigid plate in the figure.
Data: F = 10 kN; f = 5 kN/m; a = 400 mm; b = 200 mm; c = 500 mm; d = 200 mm.
F
A
b
f
a
B
d
c
2
Type 4
1,
a, Calculate the 𝐹̅𝐴 and 𝐹̅𝐵 reaction forces that act on the rod in the figure.
b, Draw the normal force (𝑁(𝑥)), shear force (𝑇(𝑥)) and bending moment (𝑀𝑏 (𝑥)) functions of the
rod applying simple rules. (You don’t have to calculate the functions here.)
Data: F1 = 20 N; F2 = 10 N; F3 = 15 N; F4 = 10 N; a = 2 m; b = 2 m; c = 4 m; d = 6 m; f = 3 N/m.
a
b
c
d
F2
f
F1
A
F4
B
F3
2,
a, Calculate the 𝐹̅𝐴 and 𝐹̅𝐵 reaction forces that act on the rod in the figure.
b, Draw the normal force (𝑁(𝑥)), shear force (𝑇(𝑥)) and bending moment (𝑀𝑏 (𝑥)) functions of the
rod applying simple rules. (You don’t have to calculate the functions here.)
Data: F1 = 6 N; F2 = 8 N; F3 = 3 N; F4 = 4 N; a = 2 m; b = 3 m; c = 2 m; d = 6 m; f = 3 N/m.
a
b
d
c
f
A
B
F2
F4
F3
F1
3,
a, Calculate the 𝐹̅𝐴 and 𝐹̅𝐵 reaction forces that act on the rod in the figure.
b, Draw the normal force (𝑁(𝑥)), shear force (𝑇(𝑥)) and bending moment (𝑀𝑏 (𝑥)) functions of the
rod applying simple rules. (You don’t have to calculate the functions here.)
Data: F1 = 9 kN; F2 = 3 kN; F3 = 6 kN; M = 1,5 kNm; f = 3 kN/m; a = 4 m; b = 2 m; c = 2 m; d = 3 m;
e = 3 m.
a
b
c
d
e
F1
F2
f
M
A
F3
B
4,
a, Calculate the 𝐹̅𝐴 and 𝐹̅𝐵 reaction forces that act on the rod in the figure.
b, Draw the normal force (𝑁(𝑥)), shear force (𝑇(𝑥)) and bending moment (𝑀𝑏 (𝑥)) functions of the
rod applying simple rules. (You don’t have to calculate the functions here.)
Data: F1 = 8 kN; F2 = 10 kN; F3 = 8 kN; M = 0,5 kNm; f = 2 kN/m; a = 4 m; b = 3 m; c = 3 m; d = 3 m; e =
2 m.
a
b
c
F2
F1
M
d
e
f
A
F3
B
9.9. ábra
5,
a, Calculate the 𝐹̅𝐴 and 𝐹̅𝐵 reaction forces that act on the rod in the figure.
b, Draw the normal force (𝑁(𝑥)), shear force (𝑇(𝑥)) and bending moment (𝑀𝑏 (𝑥)) functions of the
rod applying simple rules. (You don’t have to calculate the functions here.)
Data: F1 = 2 kN; F2 = 2 kN; F3 = 3 kN; M = 0,5 kNm; f = 1 kN/m; a = 4 m; b = 2 m; c = 2 m; d = 2 m; e =
4 m.
a
b
c
d
e
F3
F2
f
M
F1
B
A
6,
a, Calculate the 𝐹̅𝐴 reaction force and 𝑀𝐴 reaction moment that act on the rod in the figure.
b, Draw the normal force (𝑁(𝑥)), shear force (𝑇(𝑥)) and bending moment (𝑀𝑏 (𝑥)) functions of the
rod applying simple rules. (You don’t have to calculate the functions here.)
Data: F1 = 20 kN; F2 = 12 kN; f = 1 kN/m; a = 3 m; b = 3 m; c = 4 m.
a
b
c
F2
f
F1
A
7,
a, Calculate the 𝐹̅𝐴 reaction force and 𝑀𝐴 reaction moment that act on the rod in the figure.
b, Draw the normal force (𝑁(𝑥)), shear force (𝑇(𝑥)) and bending moment (𝑀𝑏 (𝑥)) functions of the
rod applying simple rules. (You don’t have to calculate the functions here.)
Data: F1 = 6 kN; F2 = 8 kN; M = 2 kNm; f = 3 kN/m; a = 4 m; b = 2 m; c = 4 m.
a
f
A
b
c
F1
M
F2
8,
a, Calculate the 𝐹̅𝐴 reaction force and 𝑀𝐴 reaction moment that act on the rod in the figure.
b, Draw the normal force (𝑁(𝑥)), shear force (𝑇(𝑥)) and bending moment (𝑀𝑏 (𝑥)) functions of the
rod applying simple rules. (You don’t have to calculate the functions here.)
Data: F1 = 6 kN; F2 = 8 kN; F3 = 30 kN; f = 2 kN/m; a = 6 m; b = 2 m; c = 4 m.
a
F1
f
b
F2
c
F3
A
Technical mechanics III
Type 1
1.
We throw a stone away from ground level. The angle between the initial velocity of the stone and
the horizontal plane is α. For the description of the motion we fix a coordinate system as it is shown
in the figure below.
y
P(x1,y1)
v0

x
m
.
2
 s 

Data: x1  100m , y1  50m ,   40 , g  9,81
Questions:
Calculate the magnitude of the initial velocity v 0 if the stone goes through point P(x1 y1).We assume
that air resistance is negligible.
2.
A child throws away a stone from ground level towards a hole. The hole is also at ground level at a
distance d from him. The magnitude of the initial velocity of the stone is v0 . The height of the child
and air resistance are negligible.
m
m
, d  125m , g  9,81 2  .

s 
s
Data: v 0  50 
Questions:
a, Calculate the angle between the initial velocity of the stone and the horizontal plane if the stone
falls into the hole.
b, Calculate the magnitude of the velocity of the stone 1s after the throw.
3.
A soldier shoots with a mortar. The magnitude of the initial velocity of the mortar bomb is
m
200  . The position of the soldier is 50m above the target, the distance between them
s
m
in horizontal direction is 500m ( g  9,81 2  ).
s 
Questions:
a, Calculate the angle between the initial velocity of the bomb and the horizontal plane if the bomb
hits the target.
(It is recommended to use the
1
2
cos 
2
 tg  1 formula for the solution.)
b, Calculate the maximum height that the bomb reaches.
c, Calculate the magnitude of the velocity of the bomb when it hits the target.
Type 2
1. A particle with mass m starts from point A from rest and goes along track ABC according
to the figure. The plane of the track is vertical, segment AB is smooth while BC is rough.
The radius R of segment AB and other data are given below.
Data:
R  10m,   60 
m  2kg ,   30 
 AB  0 ,  BC  0,1
m
h2  6m , g  9,81 2 
s 
A

R
h1
B
h2
β
C
Questions:
a, Calculate distance h1.
b, Write up the work-energy theorem on segment AB and calculate the magnitude of the velocity of
the particle in point B.
c, Calculate the work done by the gravitational and reaction forces on segment BC.
d, Write up the work-energy theorem on segment BC and calculate the magnitude of the velocity of
the particle in point C.
e, Calculate the scalar acceleration of the particle on segment BC.
f, Calculate the magnitude of the reaction force that acts on the particle on segment BC.
2. A particle with mass m starts from point A and goes along track ABC according to the
figure. The plane of the track is vertical, segment AB is rough while BC is smooth. The
radius R of segment BC and other data are given below.
Data:
s
R  2m ,   30 
 AB  0,18 ,  BC  0
m
v A  8,1 , m  5kg
s
C
B
D
α
A
vA
R

m
s  4m , g  9,81 2
s
Questions:
a, Calculate the work done by the gravitational and reaction forces on segment AB.
b, Write up the work-energy theorem on segment AB and calculate the magnitude of the velocity of
the particle in point B.
c, Write up the impulse-momentum theorem on segment AB and calculate the elapsed time between
positions A and B of the particle.
d, Write up the work-energy theorem on segment BC and calculate the magnitude of the velocity of
the particle in point C.
f, Calculate the magnitude of the reaction force that acts on the particle in point C.
3. A particle with mass m starts from point A and goes along track ABC according to the
figure. The plane of the track is vertical, segment AB is smooth while BC is rough. The
radius R of segment AB and other data are given below.
Data:
R  30m ,   60 
 AB  0 ,  BC  0,2
m
m
v A  15  , g  9,81 2 
s
s 
m
R

vA
A
vC  0
vB
C
B
Questions:
a, Calculate the magnitude of the reaction force that acts on the particle in point A.
b, Write up the work-energy theorem on segment AB and calculate the magnitude of the velocity of
the particle in point B.
c, Write up the work-energy theorem on segment BC and calculate distance s between points B and
C. (The particle stops in point C.)
d, Calculate the scalar acceleration of the particle on segment BC.
e, Calculate the magnitude of the reaction force that acts on the particle on segment BC.
Type 3
1,
The disc in the figure is moving in a horizontal plane. The velocity of point B and the radius of the disc
are given. The direction of the velocity of point A is also given in the figure.
Data:
  30  , R  0,8m
m
v B  5  .
s
vB

vA
S
B
A
R
a, Calculate the velocity of point A and the angular velocity of the disc.
b, Calculate the vector which points from point B to the instantaneous centre of zero velocity of the
disc ( r BP ).
c, Construct the instantaneous centre of zero velocity of the disc.
2,
The disc of radius R in the figure is rolling without slipping. The angular velocity of the disc and the
acceleration of its centre S are given.
Data:
y
m

s2 
R  0,5m , a S  2 
B
B
 rad 

 s 
R
R
  1
A
A
S
a
vas SS
S
C
C

x
D
a, Calculate the velocity of point A, B and C.
b, Calculate the angular acceleration of the disc and then the acceleration of point A, B and C.
c, Calculate the vector which points from point S to the instantaneous centre of zero acceleration of
the disc ( r SG ).
3,
The length and the velocity of point B of the rod in the figure are given.
Data:
B
vB
l  1,5m ,   60 
m
v B  2 
s
l

A
vA
a, Calculate the velocity of point A and the angular velocity of the rod
b, Calculate the vector which points from point B to the instantaneous centre of zero velocity of the
rod ( r BP ).
c, Construct the instantaneous centre of zero velocity of the rod.
d, Calculate the velocity of the centre of the rod and its magnitude.
4,
The length of rod AB in the figure is R 2 . The velocity of point A of it is given.
Data:
R  0,5m ,   15 
m
v A  0 ,2  
s
R
B

A vA
a, Calculate the velocity of point B and the angular velocity of the rod.
b, Calculate the vector which points from point A to the instantaneous centre of zero velocity of the
rod ( r AP ).
c, Construct the instantaneous centre of zero velocity of the rod.
5,
The mechanism in the figure is consists of three rods. The rods are attached to each other with
hinges. The length of rod AB, BC and CD are known. We also know the angles between the rods and
the horizontal plane. The angular velocity and acceleration of rod AB are given ( 1 ,  1 ).
Data:
l1  5m, l2  7m , l3  6 2 m ,
1  60 ,  2  30 ,  3  67,1
 rad 
 rad 
,  1  0 2  .
1  5 

 s 
s 
C
y
l2
B
1
A
2
l3
l1
1
3
D x
a, Calculate the velocity of point B and C and the angular velocity of rod BC and CD.
b, Calculate the acceleration of point B and C and the angular acceleration of rod BC and CD.
c, Calculate the acceleration of point B and C and the angular acceleration of rod BC and CD.
d, Calculate the vector which points from point B to the instantaneous centre of zero velocity of rod
BC ( r BP ).
e, Construct the instantaneous centre of zero velocity of rod BC.
f, Calculate the vector which points from point B to the instantaneous centre of zero acceleration of
rod BC ( r BG ).