Examples Momentum

Mechanics Examples for Topic B (Momentum) - 1 David Apsley

TOPIC B: MOMENTUM – EXAMPLES SPRING 2014

(Take g = 9.81 m s–2

).

Force-Momentum

Q1. (Meriam and Kraige)

Calculate the vertical acceleration of the 50 kg cylinder for each of the two cases illustrated.

Neglect friction and the masses of the ropes and pulleys.

Q2.

The two blocks shown start from rest. The horizontal plane and the pulley are frictionless and

the mass of the pulley and cords is negligible. Determine the acceleration of each block and

the tension in the cord.

50kg

80kg

50kg

80 x 9.81 N

(a) (b)

100 kg

300kg


Friction

Q3.

Determine the tension T in the cable which will give the 50 kg block a steady acceleration of

2 m s–2

up the incline in each case below. (Neglect the mass of cables and pulleys and assume

the pulley is smooth.)


Compute the acceleration of the block for the instant depicted. Neglect the masses of the

ropes and pulleys.

Q5.

The rod OA rotates with constant angular velocity ω = 3 rad s–1

. A small mass is placed on

the rod at a distance r = 450 mm as it passes θ = 0. The mass is observed to slip at θ = 50º.

Determine the coefficient of friction between the mass and the rod.

50 kg50 kg

= 0.25

T

30o

30o 30

o

T

(a) (b)

30o

= 0.25

40 kg

= 0.4

30o

T = 100 N

m

r

O

A



A mass m2 sits on the sloping face of a wedge-shaped block of mass m1 as shown. Neglecting

friction associated with the wheels of the tapered block, determine the range of applied force

P for which the mass m2 will not slip if the wedge angle is 20º and the coefficient of friction

between block and mass is 0.3.

Q7. (Dynamics 1, Examination May 2012 – modified)

A 200 kg mass is pulled up a 40º incline by a winch using a power of 4 kW, as shown below-

left. Initially the mass is moving with a constant speed of 2 m s–1

.

(a) Compute the coefficient of dynamic friction between the mass and the slope.

(b) Due to a malfunction in the winch, the power output is suddenly halved. Find the

acceleration of the mass immediately after this occurs.

Soon after the malfunction the cable breaks and the mass starts to fall back down the slope. It

arrives at a safety net with a speed of 12 m s–1

as shown below right. At the same moment the

net starts to impose a force P on the mass, which varies with time as the mass falls further

into the net. This variation is initially 2100tP N for the first 4 seconds, after which it

remains constant at its value after 4 s.

(c) Compute the speed 6 s after arriving at the safety net.

Impulse-Momentum

Q8.

A car with a mass of 1800 kg is driven down a 5º incline at a speed of 90 km h–1

when the

brakes are applied, causing a constant total breaking force of 6000 N. Find the time required

for the car to come to a stop. Use the impulse-momentum principle.

P

20o

= 0.3

m2

m1

winch

v

40o

200 kg

12 m/s

40oP


Collisions

Q9. (Mechanics for Civil Engineers Exam, January 2013)

Two vehicles are approaching a road junction, both moving with speed 45 kph (12.5 m s–1

).

One vehicle has mass 1000 kg and the other mass 750 kg, and the roads meet at an angle of

60º as shown. The vehicles collide and initially move as a single body.

(a) Calculate the velocity and momentum of the individual vehicles before the collision,

using the x-y coordinate system shown.

(b) Calculate the velocity of the two vehicles immediately after the collision (treating

them as a single body).

(c) After the collision, the total frictional force on the two vehicles is 6000 N, in the

opposite direction to their combined velocity. How far do the vehicles travel after the

collision?

Q10. (Mechanics for Civil Engineers Exam, January 2012 – extended)

A ball of mass 2.0 kg is travelling on a line perpendicular to a wall with speed 10 m s–1

towards a stationary ball of mass 1.5 kg, as shown. The balls are initially 5 m apart and the

stationary ball is 5 m from the wall, as shown. Assume all motion takes place on a smooth

horizontal surface, along a line perpendicular to the wall, and ignore friction. The coefficient

of restitution for collisions between the balls is e = 0.8, whilst the coefficient of resitution for

collisions between the balls and the wall is e = 0.7.

(a) Calculate the velocity and momentum of both balls after the 2.0 kg ball collides with

the 1.5 kg ball.

(b) Calculate the subsequent velocities of the balls, including all later collisions between

the balls, or between the balls and the wall.

(c) Sketch a suitable distance-time graph for the motions of the balls, indicating the times

and positions of the collisions.

12.5 m/s

1000 kg

750 kg

?

12.5 m/s

60o

x

y

5 m 5 m

2.0 kg 1.5 kg

10 m/s


Q11.

A 10 kg package drops from a chute into

a 24 kg cart with a speed of 3 m s–1

at an

angle 30º below the horizontal. If the cart

is initially at rest and can roll freely,

determine:

(a) the final velocity of the cart;

(b) the impulse exerted by the cart on

the package;

(c) the fraction of the initial energy

lost in the impact.

Q12. (Dynamics 1, Examination May 2012 – part, modified)

A ball is thrown at an angle of

70º from a height of 1 m above

floor level (point A in the

Figure). The ball follows the

trajectory shown and reaches a

maximum height at B, before

impacting the frictionless wall at

point C. It then rebounds in a

direction at tan–1

(1/3) to the

wall. Model the ball as a point

particle and assume that air

resistance is negligible.

(a) Compute the initial speed required to reach point C.

(b) Find the velocity vector just before hitting C in the x-y coordinate system shown.

(c) By considering the direction of the rebound, find the coefficient of restitution for the

impact between the wall and the ball.

Q13.

The ram of a pile driver has mass 800 kg and is released from

rest at a height of 2 m above the top of a 2400 kg pile. If the

ram rebounds to a height of 0.1 m after a direct central impact

with the pile, determine the following:

(a) the velocity of the pile immediately after impact;

(b) the coefficient of restitution;

(c) the percentage of the energy lost in the impact.

30 o

3 m/s

70o

tan = 1/3

A

B

C

6 m

3 m

1 m

x

y

2 m drop

0.1 mrebound

ram

pile



n spheres of equal mass m are suspended in

a line by wires of equal length so that the

spheres are almost touching each other. If

sphere 1 is swung aside and released and

hits sphere 2 with speed v1, find an

expression for the velocity vr of the rth

sphere immediately after being struck by the

one adjacent to it. The common coefficient

of restitution is e.


A 2 kg sphere is projected horizontally with

a velocity of 10 m s–1

against a 10 kg

carriage which is backed up by a spring with

stiffness 1600 N m–1

. The carriage is

initially at rest with the spring

uncompressed.

If the coefficient of restitution is 0.6,

calculate:

(a) the rebound velocity v;

(b) the rebound angle θ;

(c) the maximum travel δ of the carriage after impact.

Q16.

Spherical particle 1 has velocity u1 = 6 m s–1

in the direction shown and

collides obliquely with spherical particle 2 of equal mass and diameter,

which is initially at rest. If the coefficient of restitution is e = 0.6, find the

speed and direction of each particle following impact and the percentage

loss of energy in the impact.

1 2 3 4 n

v1

2 kg

10 m/s

60o

10 kg

v

k = 1600 N/m

1

2

u1


Q17. (Dynamics 1, Examination May 2011 – part (d) changed considerably)

A cannon of mass 1200 kg is located on a plateau above a gentle slope of 20º as shown. A

5 kg projectile is fired horizontally with a velocity of 200 m s–1

relative to the ground. The

projectile leaves the cannon from a point 1 m above the ground and 50 m before the start of

the slope.

(a) Apply the principle of conservation of momentum to calculate the initial recoil speed

of the cannon.

(b) If air resistance is neglected, show that the maximum range of the projectile down the

slope, s, is around 3050 m (indicated by point B) and find the time of flight. Take the

origin of coordinates to be located at the beginning of the slope.

Target T (of negligible dimensions) is moving down the slope towards point B with a

constant speed of 40 m s–1

as shown in the figure. The target has mass 200 kg.

(c) By considering the velocity of the projectile at the time calculated in part (b), show

that, at impact, the component of the projectile’s velocity acting along the slope is

around 237 m s–1

.

(d) Assuming a completely inelastic impact between the projectile and target T, compute

their combined velocity after impact.

40 m/s

50 m

1 m

s

B

T

A

20o


Centre of Mass

Q18.

Find the centre of mass of the asymmetric

rectangular frame shown.

Q19.

Where is the centre of mass of the plane

figure shown?

Q20.

(a) The uniform triangular lamina ABC shown

has weight 30 N and is right-angled at B;

AB = 300 mm and BC = 150 mm. The

lamina is suspended by vertical light

strings PA and QB and hangs in

equilibrium in a vertical plane with AB

horizontal and BC vertical (see diagram).

Find the tensions in the strings.

(b) The string QB is now cut and the lamina

settles in equilibrium supported only by the string PA. What angle does AB make

with the vertical?

Q21.

A uniform rectangular lamina has long side L and short

side 0.7 m and has a circular hole of radius 0.3 m cut

from it. The hole is equidistant from each of three

sides as shown. When hung freely from a corner near

the hole the long sides make an angle of 20° with the

vertical. Calculate the length L.

0.05 m

0.1 m

0.5 m

0.7 m

0.25 m

0.25 m

45 30o o

5 cm

16 cm

A B

C

P Q

O 0.7 m

L

0.3 m


Q22.

A uniform lamina is formed by cutting quarter-circles of

radius 0.5 m and 0.2 m from the top-right and bottom-left

corners respectively of a rectangle whose original

dimensions were 0.6 m by 0.8 m (see figure).

(a) Find the area of the lamina.

(b) Find the position of the centre of mass relative to

the top-left corner, A.

(c) If the lamina is allowed to pivot freely in a

vertical plane about corner A, find the angle made

by side AB with the vertical when hanging in

equilibrium.

(The centre of mass of a quarter-circular lamina of radius R is Rπ3

4 from either straight side.)

Q23.

A uniform solid cylinder of radius 0.05 m is held on a rough

plane inclined at 15° to the horizontal. The coefficient of friction

between plane and cylinder is 0.3. If the cylinder is then released

what happens if the height of the cylinder is:

(a) 0.4 m;

(b) 0.35 m.

A

B

0.5 m

0.2 m

0.8 m

0.6 m

0.05 m

15o

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