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A) Physical quantities
A1) Please do the following conversions of units:
i) From MKS to cgs: 300 m3 , 240 kg/m3 , 0.2 kg m
ii) from cgs to MKS: 10-5cm2 , 1,4 g2/cm3
iii) transforms 72 km/h in m/s 0,5 m/s in km/h
A2)
A3)
r r r
Three coplanar forces acting on the point P F1 , F2 , F3 with
F2
2 × 5 N . Is asked to determine
r
the intensity and direction of the force F4 that is to be applied at the
magnitudes respectively 10 N , 5 N ,
45°
F3
P
F1
point P so that the resultant of all forces is zero.
A4)
A vector A has a modulus 4 and angles (in spherical coordinates) θ=30°, φ=45°. Express
[
]
its components along the coordinate axes. Ax = 2 , Ay = 2 , Az = 2 3
r
r
r
r
A5)
Let be two vectors in a plane A ≡ (4,3) e B ≡ (3,4) . Determine the angle from A ⋅ a ⋅ B .
[16°15’ ↔ 0,28 rad]
A6)
A plane traveling from point O for 200 km west, then turns 60° to the right and continue
for 300 km to the point C. Determine the displacement final OC and its direction with respect to the
starting point. [435,88 km, 36°35’ W-NW]
A7)
A8)
A guy comes out of the door of a building 20m wide and 9m high and 10m walking
forward and 4.8 m from the side (90 °). From there throws a stone with a sling that reaches the roof
on the opposite corner to the pitcher. Knowing that the door is in the center of the facade of the
building, determine the distance traveled by the stone. (Neglecting the height and length of the arms
of the launcher!) [d=20,001 m]
A9)
A motor boat traveling at a speed Vb = 19.4 knots (= 10 m / s) to cross a river whose
current moves with velocity Vf = 3 km / h. If the river has a constant width of d = 300 m, at what
angle with respect to the shore the boat must travel along a straight section and a minimum length?
B)
B1)
Kinematics
An object travels 100 m in 20 s, then proceeds at a constant speed of 4 m / s for 30 s. What
is the total distance traveled? What is the average speed of the object along the entire route?
B2)
An object moves with accelerated motion. Three measurement are carried along its
trajectory: after 20 s to 10 m after 30 s at 20m and after 40 s at 40 m. Deduce from the graph space-
time figure in the corresponding velocity-time graph. Also calculate the average speed and average
acceleration over the entire distance traveled.
B3)
A car travels with constant acceleration and covers a distance of 58 m in 6.2 s. At the end
of the path, its speed is 15 m / s. a) What is the speed at the beginning of the route?
b) At what
distance from the beginning of the route the car was at rest? [v=3,71 m/s; x=3,78 m]
B4)
A body travels to the zero speed V0A = 10 m / s, and proceeds of uniform rectilinear
motion. After t0 = 60 s a second body B starts from standstill with acceleration aB = 2 m / s, in the
same direction and sense of A. determine:
i) In which time B reaches A;
ii) which is the distance travelled for the two bodies at the moment when they meet
What is the average speed of B in the section between its origin and the achievement of A.
[t = 90 s; sA = 900 m= sB ; v =30 m/s]
B5)
The light turns green and a car starts to move with constant acceleration a = 2.2 m/s2. At
the same time, on the lane side, comes a truck, traveling at a constant speed v = 9.5 m / s, passes the
car.
i)
How far away from the traffic lights the car to reach the truck?
ii)
what is the velocity of the car at that moment?
[x=82,1 m; v=19 m/s]
B6)
B7)
B8)
B9)
B10)
A body is dropped from a height d = 10 m. In the absence of air friction. what is the time
to touches the ground ?, and the velocity ?, if it is launched with initial speed :
i)
zero;
ii)
directed vertically upwards with module 5 m/s;
iii)
directed vertically downwards with module –5 m/s.
[t=1,43s v=14m/s; t=2,03s v=15m/s; t=1s v=15m/s]
C)
C1)
Dynamics
A body with mass m = 10 kg, subject to a force of 2.5 N intensity, runs 80 m in 5 s.
Determine the initial speed. [v0 = 15,375 m/s]
C2)
C3)
A mass m = 10 kg must be lowered by a rope with a breaking load (ie the maximum force
applicable to the rope because it does not break) equal to 70 N. what is the minimum lowering
acceleration of the mass without the rope break. [a = 2,8 m/s2]
C4)
C5)
A block of mass m = 1 kg is launched on to an inclined plane rough (µ = 0.2) with velocity
v0 = 3 m / s. If the angle of inclination is α = 30 °, calculate:
i)
the distance s traveled by the block along the plane;
ii)
the time t1 taken to cover it, as well as the total time of round trip (t1 + t2).
[s = 68,2 cm ; t1 = 0,45; t2 = 0,65]
C6)
A block of 4.8 kg on an incline of 39 ° from the horizontal is pushed with a horizontal
force of 46 N. The coefficient of kinetic friction between the block and floor is 0,33.
i)
What is the acceleration if it goes along the floor?
ii)
continuing to act the same force, how far along upward if the initial velocity is
4.3m/s?
iii)
Once reached the highest point, how it continues the block ? [a = -3,22 m/s2; s = 2,87
m; stops]
C7) An object with a mass m = 50 kg is supported by two ropes, one horizontal and the other
inclined at an angle α = 30 ° to the vertical. Calculate the tensions T1 and T2 of the two ropes under
equilibrium conditions, assuming negligible their masses. [T1 = 283 N, T2 = 566 N]
C8) A homogeneous sphere of radius r = 20 cm and density δ = 3 g /
cm is in equilibrium fixed to a wall as shown in the figure. That form
the rope angle with the wall? Determine, in addition, the tension T of
T
the rope, long l = 40 cm, and the force F that the ball exerts against the
wall.
F
[α = 19,5°, T = 1045 N, F = 349 N]
C9)
D) Uniform circular motion
D1) A point mass m = 10 kg is tied to a rope 1.5 m long and rotated evenly. In correspondence
with an angular velocity ω = 500 gpm the rope breaks. Determine the breaking load of the rope.
D2) A body with mass m = 10 kg rests on a horizontal disk 2 m from the center of the latter. If the
coefficient of static friction between the disc and the body is 0.2, what is the maximum frequency of
rotation of the disk so that the body does not drag on the disc?
D3) In the Bohr model of the hydrogen atom, the electron revolves in circular orbits around the
nucleus. If the radius of the orbit is 5.3x10-11 electron makes 6.6x1015 revolutions / s, find: (a) the
acceleration (magnitude and direction) of the electron, (b) the centripetal force acting on the
electron (this force is due to the attraction between the positively charged nucleus and the
negatively charged electron). The mass of the electron is equal to 9.1x10-31 kg. [a=9.11x1022m/s2;
F=8.29x10-8N]
E) Work and theorem of kinetic energy
E1)
A body with mass m = 8 kg is launched on a horizontal plane with a velocity of 4 m / s.
Knowing that the coefficient of friction between the ground and the body is 0.3, determine how
much space runs through the body before stopping. [2,71 m]
E2)
E3)
A bullet of mass m = 10 g, fired at a speed of 800 m / s, sticking for 20 cm inside a wall.
Determine the magnitude of the force acting between the projectile and the wall.
E4)
E5)