Download Document

AP Physics C
Dynamics
Free Response Problems
1. A block of mass m is attached to one end of a rope passing over a stationary pulley that is free to
rotate. A ring of mass M can slide on the other end of the rope. The masses of the rope and
pulley are negligible. Ignore the friction in the pulley.
a. If the block remains at rest, what is the acceleration of the ring sliding down the
rope?
b. What is the friction force between the ring and the rope?
Now with new masses, the ring moves down at a constant acceleration a with
respect to the rope.
c. What is the acceleration of the block?
d. What is the friction force between the ring and the rope?
2. A block of mass m is placed at the inclined surface with an angle θ above the horizontal. A
constant horizontal force F is applied on the block, causing the block to accelerate up the
inclined. The coefficient of kinetic friction between the block and the plane is µ. All answers
present in terms of m, θ, µ, F and g.
a. On the diagram below, show all the applied forces on the block when it moves
up the plane.
b.
c.
d.
e.
Determine the normal force applied on the block by the surface.
Determine the block’s acceleration as it moves up the inclined.
If the block slides up at a constant speed, what is the value of force F?
If the block slides down at a constant speed, what is the value of force F?
3. A horizontal force F is applied to a small block of mass m1 to make it slide along the top of a
larger block of mass m2 and length L. The coefficient of kinetic friction between the blocks is µ.
The larger block slides without friction along a horizontal surface. The blocks start from rest with
the smaller block at one end of the larger block.
a. On the diagram below draw and label all the forces acting on each block.
b. Find the acceleration of each block: a1 and a2, relative to the horizontal surface.
c. In terms of L, a1, and a2 find the time t needed for the small block to slide off the
end of the larger block.
4. Two blocks m1 and m2 are suspended at the ends of a sting that passes through a system of two
light, frictionless pulleys. The system is released from rest. (m2>m1)
a. Determine the acceleration of block m1.
b. Determine the acceleration of block m2.
c. Determine the tension force in the string.
d. Determine the support force in the cable attached to the celling.
5. A curved road with a radius of 150 m is banked at an angle of 25 ̊. The coefficient of static
friction between the tires and the surface is 0.4.
a. Find the speed of a car that doesn’t require any friction force to prevent
skidding.
b. Find the maximum speed that the car can reach before sliding up the banking.
On the diagram below show and label all the forces acting on the car at this
speed.
c. Find the minimum speed that the car can reach before sliding down the
banking. On the diagram below show and label all the forces acting on the car at
this speed.
6. A sphere of mass m is released from rest. As it falls, the air exerts a resistance force on the
sphere that is proportional to the sphere’s velocity Fr = -kv. Neglect the buoyant force of the air.
a. On the diagram below show and label all forces acting on the sphere just after it
is released.
b. On the diagram below show and label all forces acting on the sphere after it was
falling for a long time and the terminal velocity is reached.
c. Determine the terminal velocity of the sphere.
d. Draw three graphs for the sphere motion just after the sphere is released as
well as after a long time.
7. A block of mass m has an initial velocity Vo at time t = 0, slides on a horizontal surface. The
sliding friction force exerted on the block by the surface is directly proportional to its velocity Fr
= -kV.
a. Determine the amount of work that must be done on the block to bring it to
rest.
b. Determine the expression for the acceleration of the block in terms of m, k, and
Vo.
c. Determine the speed of the block as a function of time.
d. Determine the total distance the block slides.
8. A car of mass m, initially at rest at time t = 0, is driven to the right along a straight line on a
horizontal road. The engine applies a constant force Fo. While moving the car encounters a
resistance force Fr = - kv, where v is the velocity of the car and k is a positive constant.
a. On the diagram below show and label all force acting on the car as it moves to
the right.
b. Determine the horizontal acceleration of the car.
c. Derive an expression for the car’s velocity as a function of time in terms of k and
Fo.
9. A block of mass m moving along the x – axis with a velocity v is slowed by the resistance force Fr
=-kV, where k is a constant. At time t = 0, the block has a velocity Vo at position x= 0.
a. What is the initial acceleration of the block?
b. Derive an expression for the block’s velocity as a function of time t, and sketch
this function on the axes below.
c. Derive an expression for the distance of the block as a function of time and
sketch this function on the axes below.
d. Determine the traveled distance by the block form t = 0 to t =∞.
10. A block with a mass m slides down an inclined plane which makes an angle θ with the horizontal.
The block starts from rest at time t = 0 and is subject to a velocity-dependent resistance force Fr
= -bV, where V is the velocity of the block and b is a positive constant.
a. On the diagram below show and label all the applied force on the block.
b. Write but do not solve a differential equation that can be used to find the
block’s velocity.
c. Find the terminal velocity of the block.
d. Solve the differential equation for the block’s velocity and present it in terms of
m, b, g, θ.
11. A block of mass m is pulled along a rough horizontal surface by a force F that is applied at an
angle θ above the horizontal. The block moves at a constant horizontal acceleration a. Express
all the results in terms of m, θ, F, a, and fundamental constants.
a. Below show and label a free-body diagram with all forces acting on the block.
b. Write an expression for the normal force applied by the surface on the block.
c. Determine the coefficient of kinetic friction between the block and the surface.
d. Sketch two graphs: velocity and displacement as functions of time, if the block
started from rest at x = 0 and t = 0.
e. The applied force can be large enough to levitate the block above the surface.
Derive an expression for the maximum acceleration that the block still maintains
a contact with the surface.
12. A small box of mass m1 is placed on a long platform of mass m2. The system of two objects is
accelerated to the right by an applied force F. There is a maximum value of force F = Fo that the
two objects move together. The coefficient of kinetic friction between the box and the platform
is µk and there is no friction between the platform and the surface.
a. What is the acceleration of the system of two objects when the applied force is
Fo?
b. What is the acceleration of the box when the applied force F>Fo?
c. What is the acceleration of the platform when the applied force F>Fo?
d. What is the value of force Fo?
13. Two blocks with masses mB and mC are connected with a light string and placed on the surface
on an inclined plane. Block A with a mass mA is suspended from a string that goes over a pulley
and is connected to block B. The two blocks on the inclined plane move up the incline with a
constant speed. The friction force between block B and the surface is fB and the friction force
between block C and surface is fC.
a. On the diagram below show and label all forces applied on each block if mA was
moving down.
b. Find the mass of block A that the system can move with a constant speed.
c. The string between block B and C is cut, find the acceleration of block C.
14. Block B, with a mass MB, rests on the top of block A, with a mass MA, which is placed on a
horizontal tabletop. A light string attached to block A passes over a massless pulley and is
connected to block C which is suspended from the pulley. The coefficient of kinetic friction
between all the surfaces is µk and coefficient of static friction is µs.
a. Find the mass MC of block C that can prevent the two blocks from sliding on the
tabletop.
b. Find the mass MC that block C can have so the blocks will move at a constant
speed when the system is released.
c. Find the minimum value of mass Mc that will be enough to make block B to slide
off block A.
Free-Response Answers:
1.
A. 𝑎 =
(𝑀−𝑚)𝑔
𝑀
B. 𝐹𝐹𝑟 = 𝑚𝑔
C. 𝑎1 =
𝑚𝑔−𝑀(𝑔−𝑎)
𝑀+𝑚
D. 𝐹𝐹𝑟 =
2.
𝑚𝑀(2𝑔−𝑎)
𝑀+𝑚
A.
FN
F
mg
FFR
B. 𝐹𝑁 = 𝐹𝑠𝑖𝑛𝜃 + 𝑚𝑔
C. a =
𝐹(cos 𝜃−𝑢 sin 𝜃)−𝑚𝑔(𝑢 cos 𝜃+ sin 𝜃)
𝑚
D. 𝐹 =
E. 𝐹 =
𝑚𝑔𝑠𝑖𝑛𝜃+µ𝑚𝑔𝑐𝑜𝑠𝜃
𝑐𝑜𝑠𝜃
FN1
𝑚𝑔𝑠𝑖𝑛𝜃−µ𝑚𝑔𝑐𝑜𝑠𝜃
𝑐𝑜𝑠𝜃
FN1
3.
A.
FFR
F
FFR
m1g
m1g
B. 𝑎1 =
FN2
𝐹− 𝜇𝑚1 𝑔
,
𝑚1
C. t = √𝑎
2𝐿
1 − 𝑎2
m2g
𝑎2 =
𝜇𝑚1 𝑔
𝑚2
4. A. 𝑎1 = 2
B. 𝑎2 =
(2𝑚1 −𝑚2 )𝑔
4𝑚1 +𝑚2
(2𝑚1 −𝑚2 )𝑔
4𝑚1 +𝑚2
3𝑚 𝑚 𝑔
2
C. 𝑇 = 4𝑚1+𝑚
1
2
3𝑚 𝑚 𝑔
2
D. 𝑇 = 2 4𝑚1+𝑚
1
2
5. A. 26.2 m/s
B.
FN
Rg(sin 𝜃 + 𝜇 cos 𝜃)
𝑣=√
cos 𝜃 − 𝜇 sin 𝜃
FFR
V=39.6 m/s
mg
FN
C.
FFR
Rg(sin 𝜃 − 𝜇 cos 𝜃)
𝑣=√
cos 𝜃 + 𝜇 sin 𝜃
Fr
6.
A.
mg
Fr
B.
mg
mg
V=9.1m/s
𝑚𝑔
𝑘
C.
D.
7.
A.
1
𝑚𝑣𝑜 2
2
B. 𝑎 =
−𝑘𝑣
𝑚
C. v = 𝑣𝑜 ∙ 𝑒
D. x=
−𝑘𝑡
𝑚
𝑣𝑜 𝑚
𝑘
FN
8. A.
FR
Fo
mg
B.
𝑎=
𝐹𝑜 − 𝑘𝑣
𝑚
C.
v=
𝐹𝑜
𝑘
−𝑘
(1 − 𝑒 𝑚 𝑡 )
9.
A. 𝑎 =
−𝑘𝑣
𝑚
B. v = 𝑣𝑜 ∙ 𝑒
10.
C. x =
𝑣𝑜 𝑚
𝑘
D. 𝑥 =
𝑚𝑣𝑜
𝑘
−𝑘𝑡
𝑚
−𝑘
(1 - 𝑒 𝑚 𝑡 )
A.
FN
mg
B.
𝑚𝑔 sin 𝜃−𝑏𝑉
𝑚
=
𝑑𝑣
𝑑𝑡
𝑚𝑔 sin 𝜃
𝑏
C. 𝑣𝑡 =
D. 𝑣 =
11.
𝑚𝑔 sin 𝜃
𝑏
−𝑏
(1 − 𝑒 𝑚 𝑡 )
FN
A.
mg
B. −𝐹 sin 𝜃 + 𝑚𝑔 = 𝐹𝑁
C.
𝐹 cos 𝜃− 𝑚𝑎
−𝐹 sin 𝜃+𝑚𝑔
= 𝜇𝑘
D.
E.
𝐹 cos 𝜃
𝑚
12.
A. 𝑎 = 𝑚
𝐹𝑜
1 +𝑚1
B. 𝑎1 =
𝐹𝑜 −µ𝑘 𝑚1 𝑔
𝑚1
C. 𝑎2 =
µ𝑘 𝑚 2 𝑔
𝑚2
D. 𝐹𝑜 =
µ𝑘 𝑚1 𝑔(𝑚1 +𝑚2 )
𝑚2
F
13.
A.
FNB
FT1
FT1
FT2 FT2
fB
mB g
mAg
B. 𝑚𝐴 =
C. 𝑎 =
14.
mcg
(𝑚𝐶 + 𝑚𝐵 )𝑔 sin 𝜃+ 𝑓𝐶 + 𝑓𝐵
𝑔
𝑚𝐶 𝑔 sin 𝜃− 𝑓𝐶
𝑚𝐶
A. 𝑀𝑐 = 𝜇𝑠 (𝑀𝐴 + 𝑀𝐵 )
B. 𝑀𝑐 = (𝑀𝐴 + 𝑀𝐵 )𝜇𝑘
C.
FNC
2𝜇(𝑚𝑎 +𝑚𝑏 )
1−𝜇
< 𝑚𝑐
fC