Dynamics (EPHS 120) Summer 2016
SHEET (1)
1. The motion of a particle is defined by the relation x = t3−9t2+24t−6, where x is expressed in
meters and t in seconds. Determine: the position, velocity and acceleration when t = 10 s.
2. The motion of a particle is defined by the relation x = t3−12t2+36t+30, where x is expressed
in meters and t in seconds. Determine the position and acceleration when v = 0.
3. The acceleration of a particle is defined by the relation a = −𝒌𝒙−𝟐 . The particle starts with no
initial velocity at x = 800 mm, and it is observed that its velocity is 6 m/s when x = 500 mm,
determine:
(a) The value of k.
(b) The velocity of the particle when x = 250 mm.
4. The acceleration of a particle is defined by the relation a = −k/x. It has been experimentally
determined that v = 14 m/s when x = 0.4 m and that v = 8 m/s when x = 0.8 m. Determine:
(a) The velocity of the particle when x = 1 m.
(b) The position of the particle at which its velocity is zero.
5. The acceleration of a particle is defined by the relation a = −𝒌𝒗𝟐.𝟓 , where k is a constant. The
particle starts at x = 0 with a velocity of 16 mm/s, and when x = 6 mm the velocity is observed to
be 4 mm/s. Determine:
(a) The velocity of the particle when x = 5mm.
(b) The time at which the velocity of the particle is 9 mm/s.
6. Determine the height h on the wall to which the
firefighter can project water from the hose.
( vc = 48 m/s, h1 = 1 m, d = 30 m, θ = 40°, g = 9.81 m/s)2.
Summer 2016 Page 1 of 2
7. An airplane used to drop water on brushfires is
flying horizontally in a straight line at 315 km/h at an
altitude of 80 m. Determine the distance d at which
the pilot should release the water so that it will hit the
fire at B.
8. The man stands at a distance d from the wall and throws a ball at it with a speed v0. Determine
the angle θ at which he should release the ball so that it strikes the wall at the highest point
possible. What is this height? The room has a ceiling height h2.
(d = 20 m v0 = 15 m/s h1 = 1.5 m h2 = 6 m g = 9.81 m/s2)
9. Milk is poured into a glass of height 140 mm and
inside diameter 66 mm. If the initial velocity of the milk
is 1.2 m/s at an angle of 40° with the horizontal,
determine the range of values of the height h for which
the milk will enter the glass.
Summer 2016 Page 2 of 2
Dynamics (EPHS 120) Summer 2016
SHEET (2)
1. The crane lifts a 700 kg bin with an acceleration of 3 m/s2. Determine the force in each cable
due to this motion, knowing that b = 3, c = 4.
2. A 500-kg mass is being lifted out using the pulley system shown with upward acceleration.
Initial values are s = 0 and v = 0 when t = 0 and s = 2.5 m when t = 1.5 s. Determine the tension
in the cable.
3. Each of the two blocks has a mass m. The coefficient of
kinetic friction at all surfaces of contact is µ. If a horizontal
force P moves the bottom block, determine the acceleration
of the bottom block in each case.
Summer 2016 Page 1 of 2
4. The acceleration of a package sliding at point (A) is 3 m/s2. Assuming that the coefficient of
kinetic friction is the same for each section (µk is constant), determine the acceleration of the
package at point (B).
5. A light train made up of two cars is travelling at 88 km/hr when the brakes are applied to both
cars. Knowing that car (A) has a mass of 25000 kg and car (B) has a mass of 20000 kg and that
the braking force is 31000 N on each car, determine:
(a) The distance travelled by the train before it comes to stop.
(b) The force in the coupling between the cars while the train is slowing down.
6. Neglecting axle friction and the masses of the pulleys, determine:
(a) the acceleration of block A, (b) the velocity of block A after it has moved
through 3m, (c) the time required for block A to reach a velocity of 6 m/s.
Summer 2016 Page 2 of 2
Dynamics (EPHS 120) Summer 2016
SHEET (3)
.
1. A 100 kg crate is subjected to forces F1 = 800 N and F2 = 1500 N, as shown. If it is originally
at rest, determine the distance required to attain a speed of 6 m/s. Given that θ1 = 30°, θ2 = 20°
and µk = 0.2.
2. A 4 kg stone is dropped from a height h and strikes the ground with a velocity of 25 m/s. Find
the kinetic energy of the stone as it strikes the ground and the height h from which it was
dropped.
3. The conveyor belt delivers crate each of 12 kg to the ramp at A such that the crate’s velocity
is 2.5 m/s directed down along the ramp. If the coefficient of kinetic friction between each crate
and the ramp is 0.3. Determine the speed at which each crate slides off the ramp at B. Assume
that no tipping occurs. Given that: a = 3 m and θ = 30°.
4. A spring AB of constant k is attached to a support at
A and to a collar of mass m. The unstretched length of
the spring is l. Knowing that the collar is released from
rest at x = x0 and neglecting friction between the collar
and the horizontal rod, determine the magnitude of the
velocity of the collar as it passes through point C.
(Hint: use conservation of energy)
Summer 2016 Page 1 of 2
5. Packages are thrown down an incline at A with a velocity of 1.5 m/s. The packages slide along
the surface ABC to a conveyor belt which moves with a velocity of 3 m/s. Knowing that µk =
0.35 between the packages and the surface ABC, determine the distance d if the packages are to
arrive at C with a velocity of 3 m/s.
6. The spring has a stiffness of 1000 N/m and an un-stretched length of 1 m. For the assembly
shown, l = 0.7 m, μk = 0.2, a = 3, b = 4, d = 2 m. A 2- kg block strikes the spring and pushes it
forward a distance of 0.1 m before stopping, determine the speed at A.
Summer 2016 Page 2 of 2
Dynamics (EPHS 120) Summer 2016
SHEET (4)
1. The disk of radius 0.5 m is originally rotating at angular velocity of 8 rad/s.
If it is subjected to a constant angular acceleration of 6 rad/s2, determine the
magnitudes of the velocity and the n and t components of acceleration of
point A at the instant t = 0.5 s.
2. A motor gives gear A angular acceleration αA = 4t3. If this gear is initially
turning with angular velocity 20 rad/s, determine the angular velocity of gear
B when t =2 s. (rA = rB/3).
3. Ring C has an inside radius of 45 mm and an outside radius of 50
mm and is positioned between two wheels A and B, each of 24-mm
outside radius. Knowing that wheel A rotates with a constant angular
velocity of 400 rpm and that no slipping occurs, determine (a) the
angular velocity of the ring C and of wheel B, (b) the acceleration of
the points A and B which are in contact with C.
4. Disk A has a clockwise angular acceleration αA = 0.6t2 + 0.75. If the initial
angular velocity of the disk is 6 rad/s, determine the magnitudes of the velocity and
acceleration of block B when t = 2 s. (rA = 0.15 m).
Summer 2016 Page 1 of 2
5. Bar AB is moves along the vertical and inclined planes. If the velocity of the roller at A is 6 m/s
downward when θ = 45⁰, determine the bar's angular velocity and the velocity of roller B at the instant
φ= 30°, L = 2 m .
6. Collar A moves upward with a constant velocity of 1.2 m/s. At the instant shown when θ = 25°,
determine:
The angular velocity of rod AB.
The velocity of collar B.
Summer 2016 Page 2 of 2
Dynamics (EPHS 120) Summer 2016
SHEET (5)
1. Determine the maximum possible acceleration of the rear wheel drive car shown, car mass is 975 kg,
static coefficient of friction is 0.8, kinetic coefficient of friction is 0.6, a = 1.82 m, b = 2.2 m and h =
0.55 m.
2. Resolve the previous problem assuming the car is four wheel drive.
3. The crate of mass m is supported by a cart of negligible mass. Determine the maximum force P that
can be applied a distance d from the cart bottom without causing the crate to tip on the cart.
4. The coefficient of static friction between the wheel and the road is 0.8. Knowing that the total mass is
80 kg, a = 0.5 m, b = 0.4 m and c = 1.2 m, determine the normal reactions at the tires A and B, and the
deceleration when the brakes cause the rear wheel to be locked.
Page 1 of 1
