# KINEMATICS IN TWO-DIMENSIONS by liuhongmei

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```									                                Chapter 5: NEWTON’S LAWS II:
Application of Newton’s Laws

When solving problems with Newton’s 2nd Law keep in mind that the correct free-body-diagrams
are essential, and that you should always choose your coordinate system to be parallel to the
direction of acceleration. Remember that only real interactions can apply forces on objects.

Each “fbd” should give you two working equations that can be combined to determine the answer
to the problem. You can also use kinematics formulas to find accelerations.

1. There are some common misconceptions about the normal force that deserve consideration.
Answer TRUE or FALSE to the following statements and justify your decision.

a) The normal force is not always equal to the weight.
b) The normal force is always vertical.
c) The normal force can accelerate an object.

2. The force of friction can be difficult to deal with. Answer TRUE or FALSE to the following

a)   Kinetic friction is generally less than static friction.
b)   Kinetic friction is always equal to the coefficient of kinetic friction times the normal force.
c)   Static friction is always equal to the coefficient of static friction times the normal force.
d)   Friction always slows down the speed of an object so it cannot accelerate an object.
e)   Friction can make an object turn around.
f)   Friction cannot exist on a vertical surface.
g)   The coefficients of friction are measured in Newtons.

3. In order to solve problems in circular motion it is important to understand the centripetal
force. Answer TRUE or FALSE to the following statements and justify your decision.

a) The centripetal force is an extra force on a body generated by the circular motion.
b) The centripetal force is simply another name for the net force on a rotating body directed
c) In a rotating frame of reference an object in circular motion experiences a sense of being
pulled outward by a (centrifugal) force, but this force does not exist in an unaccelerated
(inertial) frame of reference.
d) The centripetal force can be any real force (such as weight, tension, friction, etc) or
combination of real forces.
e) Static friction cannot be a centripetal force because the object is in circular motion.
f) One should never draw the centripetal force in a free-body diagram.

4. In the following cases draw the “fbd” for the highlighted object paying special attention to the
force of friction and especially to its direction. For each case determine whether the force of
friction is accelerating, decelerating or balancing the other forces. Determine a valid
expression for the force of friction on each. You may assume that the coefficients of friction
are known.
Illustration               Free-body diagram      Acceleration,     Force of friction         Net force
deceleration      and its direction      on object and its
or neither?                                direction.
Box of mass M at rest relative
to an accelerating cart.
a

Book of mass M is held at rest
on a wall by a
horizontal force F

A car rounding a curve on the

Block of mass M slides up a
rough ramp of angle 
vo

5. A student sets up an experiment to determine the coefficient of friction between a block and a long
board. He places the block on the board then he slowly tilts the board up, increasing the angle until the
block just begins to slide over the board.

a) What does the student need to measure to determine the coefficient of                             
friction between the block and the surface? Does this give him/her the
coefficient of static friction or of kinetic friction?
b) Would the result in (a) change if: (i) the block was heavier? or if (ii) the contact surfaces were
larger?
c) When the block begins to slide it will actually accelerate downward (if no change is made on
the angle). Explain why. Determine the acceleration.
d) How could the student determine the coefficient of kinetic friction?

6. Consider the problem of a mass on an incline connected to another mass over a pulley. We
have done some versions of this problem in class, but there are many variations. We used an
angle of 30o for the incline, and the masses were 0.3 kg and 0.2 kg. In class we showed that, with
no friction, the acceleration of the system was 1 m/s2 with the hanging mass moving downward.
a
a) Suppose you replace the hanging mass with a different
one and that, as a result, the acceleration changes to 1 m/s2
in the opposite direction. What is the value of the mass and                   
tension in the rope if we assume that no friction is present?
b) Determine the force of friction that would keep the original system of the 0.3 kg and 0.2
kg at rest. What minimum coefficient of static friction does this require?
c) Assume the system at rest in (b) is jolted temporarily and, as a result, the masses begin to
accelerate. The tension drops to 1.9 N in that case. Determine the acceleration and the
coefficient of kinetic friction.
d) Determine the angle that would keep the system at rest with no help from friction.
e) Determine the acceleration in two extreme cases: (i) the angle of the incline is zero, and
(ii) the angle of the incline is 90o.

7. Two masses 2-kg each and two identical springs are arranged in three different ways. In Fig. 1
below the lower spring is stretched 5 cm. Assume massless springs.

a) Show that the spring constant is 400 N/m and that the other spring in Fig.1 is stretched 10 cm.
b) Determine the amount of stretch if the entire system accelerates downward at 3 m/s2.
c) Now consider the other two arrangements of these same elements. Determine the stretches in
each spring in each case.
d) A spring is considered “stiffer” than another spring if it exerts more force with less deformation
(this means the spring constant “k” is larger). Combining two (or more) springs can increase or
decrease the overall stiffness of the combination. In Fig. 2 the springs are acting “in parallel”, while
in Fig. 3 the springs are acting “in series”. In which case is the overall stiffness greater than that of a

Fig. 1                              Fig. 2                                Fig. 3

8. Double pulley systems can give you what is called “advantage”. For the following cases,
determine the relationship between the tension in the difference strings, and the relationship
between acceleration of the masses involved.

Illustration          Tension                          Acceleration            Acceleration in terms
Relationships                      Relationship            of the masses and “g”

m1
m2

M

m
9. Consider three versions of a problem with three block of masses m, 4m, and 3m. The small mass is
always on top of the large mass, and the 3m mass is accelerating downwards (but at different rates).
There is friction between the m and 4m block but not anywhere else. The coefficients of kinetic and
static friction are 0.6 and 0.9 respectively. Pulleys and strings are massless.

m

4m

a1
a2                                  a3
3m

a) Draw the forces of friction in each case. Which are static and which are kinetic?
b) In which case is the acceleration the greatest? …the least? Justify your choice.
c) Determine the accelerations in each case and check your answer to (b).

10. A 2-kg box sits on a long platform of mass 3-kg and length 80 cm. There is friction between the
box and the platform (k=0.3 and s= 0.4). There
is no friction between the platform and the ground.                                               F

a) A horizontal force F is applied to the platform so that both objects accelerate together and the
box stays at rest relative to the platform. Determine the maximum possible acceleration and the
value of F in this situation.
b) Draw the “fbd” for the box. What is the accelerating force on the box? What function does the
force of friction serve in this situation? Is it static or kinetic friction that is involved here?
c) Draw the “fbd” for the platform. What is the accelerating force on the platform? Compare the
force of friction on the platform to the force of friction on the box.
d) A temporary jolt causes the box to break the static contact and to begin to slide relative to the
platform. Determine the acceleration of the box relative to the ground, and the acceleration of
the box relative to the platform. Make sure to give directions as well as magnitudes.

11. Here is another version of the problem above. A 2-kg box sits on a long platform of mass 3-kg and
length 80 cm. There is friction between the box and the platform (k=0.3 and s= 0.4). There is no
friction between the platform and the ground. In this                                  F
version the horizontal force F is applied directly to the box.

a) Assume that both objects accelerate together and that the box stays at rest relative to the
platform. Determine the maximum possible acceleration and the value of the force F in this
situation.
b) Is the resultant acceleration smaller or larger than in the previous problem? Describe the
differences between these two problems.
c) The force F is increased to 20 N and the box then slides relative to the platform. Determine the
acceleration of the box relative to the ground, and the acceleration of the box relative to the
platform. Make sure to give directions as well as magnitudes.
d) In (c) the box is dragged over the full length of the platform. How far did it actually move
relative to the ground? (Hint: you need to find the time it took the box to slide from one end of
the platform to the other.)

12. In the different cases below determine the force F needed to move a mass M at constant speed
(either up or down) on an incline of angle with a coefficient of kinetic friction k.
(a)                            (b)                               (c)                     (d)

V constant
V constant                     V constant                        V constant

F                                                                 F                              F
                                    F
                                                              

13. A box is given an initial speed vo up an incline of angle . The coefficients of kinetic and static
questions below in terms of the given terms.
vo

a) Show that the ratio of the acceleration of the box going up the ramp to                       
the acceleration of the box going down the ramp is:
aup/adown = (tan+ k)/(tan - k).
b) Assume that the box goes up and that it immediately comes down. Determine the final speed
when it returns to the original position.
c) Draw a velocity vs. time graph for (b).
d) Explain why it takes longer for the box to come down than it takes for it to go up. Determine
the ratio of the time up to the ratio of the time down.
e) If the angle is sufficiently large the box will reach the top but it will not come down.
Determine that minimum angle.

14. A person is pulling on a crate of mass M at an angle. The crate is dragged at constant speed
over the floor where the coefficient of kinetic friction is k.           F(d)
F(a,b,c)
                
a) Determine the value of F if the angle were zero.
b) If the angle is increased from zero, does it get easier or
c) At what angle does it take the least effort to drag the crate? (This problem requires calculus to
find the minimum value of a function.)
d) Instead of pulling the crate the person decides to try to push on it from above at an angle .
What is the maximum angle above which no amount of pushing will move the crate?
e) What is the value of k that would make it impossible to move the crate for a given push and
angle?

15. A box of mass m sits on a wedge of mass M and angle . A horizontal force F is applied to the
wedge. Assume all surfaces are frictionless. The wedge and box are accelerating horizontally such that
the box remains at rest relative to the wedge. In terms of the given quantities determine the following.
(Hint: Note the direction of the acceleration!)
a
a) Draw a “fbd” for the box. What’s the accelerating force on the box?             F


b) What is the acceleration “a” in this situation? What happens if the
acceleration of the wedge is greater (or lesser) than the “a” derived above?
c) What is the value of the force F required to maintain the acceleration “a”?
d) If there is some friction between the wedge and the box (s for example), there is a range of
accelerations for which the box can remain at rest relative to the wedge with the help of static
friction. Determine the maximum and minimum accelerations of this range.                       a
o
e) Consider this problem in the extreme case of the angle going to 90 . What minimum
acceleration is required to keep the box at rest relative to the surface?              F

16. In the following cases of circular motion determine a valid expression for the centripetal force and
for the tangential force (if any) in terms of the actual forces on the highlighted body. Remember that
centripetal and tangential forces are net forces on a rotating body.

Illustration                      Free-body diagram                   Centripetal force and
tangential force
Conical pendulum of
length L, mass m, and
angle 
Satellite in orbit
near the earth surface

Swinging pendulum at the
bottom and at the ends      

A man standing on a
rotating turntable

A jogger running of a
banked track of
angle 

17. A bug is crawling toward the rim of a turntable along a radius. The turntable is rotating at a rate of
0.5 rps (revolutions/sec) and it has a radius of 15 cm.

a) When the bug is 10 cm from the center, it begins to slip on the surface of the turntable. What is
the coefficient of static friction between the turntable and the bug?
b) As the bug crawls radially outward, does the time to go around one revolution increase,
decrease, or stay the same? Does its speed increase, decrease, or stay the same? Does its
c) How slow would the turntable have to rotate so that the bug can reach the rim of the turntable
without slipping?

18. Consider a car going around a corner on a level road. Assume that the mass of the car is M, the
coefficient of friction is  and the radius of curvature is R.
a) Draw the “fbd” on the car and discuss the role of the force of friction on the car. Is the force of
friction static or kinetic? What would happen if the car hit a slippery part of the road?
b) Determine the maximum speed with which the car can safely round the corner. Is there a
minimum speed? Explain.
c) Suppose that the radius of curvature of the road is 25 m and that the posted speed limit is 20
mph (about 10 m/s). What minimum coefficient of friction is implied by that information?
d) What would be the difference if the car were loaded with people instead of just carrying the
driver?

19. Consider a car going around a corner on a road banked at an angle . Assume that the mass of the
car is M, the coefficient of static friction is  and the radius of curvature is R. Some typical values
would be R=90 m, = 35o, and s=0.5.

Cross-section of

R

a) Show that if the car’s speed is equal to (Rgtan)1/2 the car can safely round the curve without
generating any friction. (Hint: Draw the “fbd” and pay attention to the direction of acceleration
of the car).
b) What would be the range of safe speeds (vmax and vmin) in the banked road if friction were
taken into consideration? (Hint: Add the force of friction to the “fbd” above and note that its
direction can be either up or down the incline.)
c) Consider the extreme case of a 90o-banked curve.
This amounts to a vertical cylindrical wall. Can a car
drive around the inside of a cylindrical wall? How fast
would it have to travel to keep from sliding down the wall?

20. Amusement parks have many examples of circular motion. Consider the “loop-the-loop”. This is a
vertical track that allows a passenger to ride a cart upside down. Assume that the track is frictionless
and that the radius of the track is R. Consider the cart of mass M at different locations on the track.
Determine the centripetal force in terms of the real forces on the cart. (Hint: This problem is nearly
identical to a pendulum making a full vertical swing.)

a)
Illustration                                       v         v                 v

g                                                         v
g                                                 g
g                g
v
Centripetal
force

Does
passenger
feel lighter
or heavier?
b) Determine the minimum speed that the cart can have at the top of the ride.
c) Determine the speed of the cart if it loses contact (N =0) with the track before it reaches the top.
d) For safety’s sake the “loop-the-loop” is actually made up of two circular
tracks where the cart’s wheels ride. Explain how a double track keeps the cart
from leaving the circle regardless of speed.

21. There are many other interesting amusement park rides. Determine the centripetal force in each of
the following.

a) Determine centripetal forces in terms of the actual forces on the rider.
Roller coaster                   Ferris wheel                Spinout
Illustration                              g                                                              g
g

R

Fc at the top                                                                            x
Fc at the bottom                                                                          x
Fc at the middle                      x

b) Why is it unsafe to go too fast over the top of a roller coaster? Determine the maximum safe
speed in terms of radius of curvature on the track and “g”.
c) Describe how a person “feels” riding the Ferris wheel at different locations.
d) What is the minimum rotating speed in the Spinout that would allow a person to “stick to the
wall” while the floor is lowered?

22. A 5-kg mass is attached to a 50 cm long string. A spring of constant 200 N/m is also attached to
the mass. The other end of the string and the spring are attached to a stick and set 40 cm apart. The
stick is rotated at different rates about its length making the spring compress or stretch. This forces the
string to take on different positions and tensions. Three situations are illustrated below. In the first
illustration the spring is compressed 10 cm, in the second the spring is relaxed, and in the third the
spring is stretched 10 cm. For each situation determine the tension in the string, the speed of the mass,
and the rate of rotation or frequency (in rps) of the system.

50 cm                                                                cm
50 cm
40 cm        
                        40 cm                                   40 cm

cm
30 cm                        cm

23. A small ball with a mass of 400 g is attached to a stick with two strings A and B of equal length
of 50 cm. The strings are attached 80 cm apart on the stick. The stick rotates at a rate of 2 rps keeping
the strings extended and taut.
a) Determine the tension in each string.
b) Slow down the rotation until the lower string is barely taut.      
How slow is this rotation?
c) Repeat the problem for the case where the stick is rotating
about a horizontal axis. Determine the tensions when the
ball is at its highest point and when it is at its lowest point.



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