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Physics 6A
Practice Midterm #2
solutions
1. A locomotive engine of mass M is attached to 5 train cars, each of mass M. The engine produces
a constant force that moves the train forward at acceleration a. If 3 of the cars are removed, what will
be the acceleration of the shorter train?
We just use Newton’s 2nd law here: Net force = (total mass) x (acceleration)
Initially, the total mass is 6M (5 cars, plus the engine)
After removing 3 cars the total mass is 3M (2 cars, plus the engine)
In both cases the Net force is the same (the engine didn’t change).
Here is the formula in both cases:
Initial
Final
F = (6M)(a)
F = (3M)(afinal)
Setting these equal gives afinal = 2a
Another way to think about this one is that since the mass is cut in half, the acceleration must double.
2. Two boxes are placed next to each other on a smooth flat surface. Box A has mass 1 kg and Box B
has mass 3 kg. A constant horizontal force of 8 N is applied to Box A. Find the force exerted on Box B.
We can think of this as a single box with total mass 4kg.
Then using F = ma we get the acceleration of the whole system.
8N = 4kg ⋅ asystem
B
asystem = 2 m2
A
s
Now we do the same thing, but just for box B:
FB = 3kg ⋅ 2 m2
s
FB = 6N
Remember – Box B has a mass of only 3kg
3. Two boxes are placed next to each other on a flat surface. Box A has mass 1kg and Box B has
mass 3kg. The coefficients of friction are 0.3 and 0.4 for kinetic and static friction, respectively. A
constant horizontal force of 8N is applied. Find the acceleration of Box B.
We can think of this as a single box with total mass 4kg.
Then using F = ma we get the acceleration of the whole system.
We need to account for friction, so first find the maximum force of static friction:
Fstatic ,max
= μs ⋅ N
Fstatic ,max
= (0.4 ) ⋅ 4kg ⋅ 9.8 m2 = 15.68N
(
s
)
8N
B
A
Fstatic
This friction force is more than the 8N force trying to move the boxes, so they are held in
place by friction. Thus the acceleration of both boxes is 0.
*Note that the actual force of friction holding the boxes in place is only 8N – just enough to
keep them from moving.
4. A crate filled with books rests on a horizontal floor. The total weight of the crate and books is 700N.
The coefficients of friction are 0.35 for kinetic and 0.45 for static. A force of 450N is applied to the
crate at an angle of 20° below the horizontal. Find the acceleration of the crate.
We need to find the components of the
450N force:
F x = 450
N
F y = 450
N
(20 ) ≈ 423
⋅ sin (20 ) ≈ 154
o
⋅ cos
o
N
FNormal
Ffriction
Fx
N
We need to find the normal force, so we can determine whether
static friction will hold the crate in place. For this, we use the fact
that the forces in the y-direction cancel out:
Fnet,y = Fnormal − 700N − 154N = 0
Fy
700N
Fnormal = 854N
This gives a maximum force of static friction = (0.45)(854N) = 384N
Comparing this with the x-component of the applied force, we see that this is not enough
friction to hold the crate in place, so the actual friction force will be kinetic.
Ffriction,k = (0.35)(854N) = 299N
Now we can use F=ma again for the x-direction forces:
Fnet ,x = Fx − Ffriction,k
⎛ 700N ⎞
⎟ ⋅ a ⇒ a = 1.74 m
= ma ⇒ 423N − 299N = ⎜
m
⎜ 9 .8 ⎟
s2
2
s ⎠
⎝
5. Blocks A and B and C are connected by a massless string and placed as shown, with Block A
on the upward slope of the 60° incline, Block B on the horizontal surface at the top and Block C
on the downward slope of the 60° incline. The string passes over a frictionless, massless pulley.
The inclines are frictionless, but the horizontal surface has coefficient of kinetic friction 0.45.
Blocks A and B have mass 5kg. The system accelerates at 2 m/s2. Find the mass of Block C.
NB
Fk
a=2 m/s2
B
T2
T1
T1
NA
mBg
T2
A
C
NC
mAgcosθ
mCgcosθ
mAgsinθ
mAg
mCgsinθ
mCg
Here is the force diagram for this problem. The blocks will move together, so we can consider the motion of
the entire system in one formula. Our axis system will be chosen to coincide with the string.
With this choice, our formula is: − m A g sin θ − Fk + mCg sin θ = (mA + mB + mC ) ⋅ asystem
Notice that the tension forces cancel out. We are assuming the string and pulley are massless, so the
tension is the same throughout the string. Thus the forward tension T1 on A is canceled by the backward
tension T1 on B, and similarly the T2 cancels out. Or we say that the tensions are internal forces, and thus
do not affect the acceleration of the entire system.
5. Blocks A and B and C are connected by a massless string and placed as shown, with Block A
on the upward slope of the 60° incline, Block B on the horizontal surface at the top and Block C
on the downward slope of the 60° incline. The string passes over a frictionless, massless pulley.
The inclines are frictionless, but the horizontal surface has coefficient of kinetic friction 0.45.
Blocks A and B have mass 5kg. The system accelerates at 2 m/s2. Find the mass of Block C.
− mA g sin θ − Fk + mCg sin θ = (mA + mB + mC ) ⋅ asystem
Now we need to find the friction force. For this we need the Normal force on B.
From the diagram we see that NB = mBg.
Since the blocks are in motion, we are dealing with kinetic friction.
(
)
Fk = μk ⋅ mBg = (0.45 ) 5kg ⋅ 9.8 m2 = 22.05N
s
Now we can substitute into the force formula:
(
) ( )
(
) ( )
( )
− (5kg) 9.8 m2 sin 60o − (22.05N) + mC 9.8 m2 sin 60o = (5kg + 5kg + mC ) ⋅ 2 m2
s
s
A bit of algebra will get us the desired mass mC
− 64.5N + mC (8.5 ) = 20N + (2)mC
(6.5)mC = 84.5N
mC = 13kg
s
6. A 0.2 kg piece of wood is being held in place against a vertical wall by a horizontal force of 5 N.
Find the magnitude of the friction force acting on the wood.
From the force diagram, we can see that the
friction force must equal the weight of the piece
of wood to keep it from falling.
Ffriction = mg
(
Ffriction = (0.2kg) 9.8 m2
Ffriction = 1.96N
s
Ffriction
)
N
5N
mg
7. A 4 kg block on a horizontal surface is attached to a spring with a force constant of 50 N/m. As
the spring and block are pulled forward at constant speed, the spring stretches by 25 cm. Find the
coefficient of kinetic friction between the block and the table.
The key phrase here is “constant speed”. Since the block is moving at constant speed (and
direction) we know that its acceleration is 0. Thus the net force is 0 as well.
Here is the force diagram. Since the net
force is 0, we know that the Normal force
must equal the weight, and the Friction
Ffriction
force must equal the Spring force.
The spring force obeys Hooke’s Law:
Fspring = k ⋅ x =
(
N
50 m
)(0.25m) = 12.5N
The weight is:
mg = (4kg) 9.8 m2 = 39.2N
(
s
)
Finally, we can put this together to find the friction:
Fspring = Ffriction = μk ⋅ N
12.5N = μk ⋅ (39.2N)
μk = 0.32
N
Fspring
mg
The friction force must be directed toward the
center of the circle. Otherwise the car will slide off
the road. If we want the maximum speed, then we
want the maximum static friction force. The road is
flat (not banked), so the Normal force on the car is
just its weight.
(
Ffriction
50 m
8. A 1000 kg car is driven around a turn of radius 50 m. What is the maximum safe speed of the car if
the coefficient of static friction between the tires and the road is 0.75?
)
Ffriction,static,max = μs ⋅ mg = (0.75 )(1000kg) 9.8 m2 = 7350N
s
This friction force is the only force directed toward the center, so it must be the centripetal force:
Ffriction = Fcentripetal
mv 2
=
r
(
1000kg)(v 2 )
7350N =
50m
v
= 19.2 ms
9. A 90 kg man drives his car at a constant speed of 15 m/s over a small hill that has a circular cross
section of radius 40 m. Find his apparent weight as he crests the top of the hill. (Hint: the apparent weight
is the same as the normal force on the man.)
15 m/s
When the car reaches the top of the hill, it will have
only 2 forces: its weight, and the normal force
supplied by the road. Since the man is sitting in the
car, he feels a normal force as well.
The car (and man) must be accelerating toward the
center of the circle, so the net force on the man will
be equal to the centripetal force required to keep
him moving along the circle.
∑ Fcent = mg − Normal =
mv
r
mv 2
Normal = mg −
r
(
)
Normal = (90kg) 9.8 m2 −
Normal = 376N
s
2
(90kg)(15 ms )2
40m
40 m
N
mg
10. Planet X has a radius that is 3 times as large as Earth and a mass that is 6 times that of Earth. A NASA
astronaut who weighs 550 N here on Earth is planning to embark on a manned mission to Planet X. What
will be the astronaut’s weight when she lands there?
The weight is the same as the gravitational force. On
Earth the weight is 550N:
Fgrav =
G ⋅ MEarth ⋅ m
(rEarth )2
= 550N
In this formula, G is constant, and m will not change when
the astronaut goes to a new planet. So the only parts that
will change are the mass and radius of the planet.
The easy way to do this problem is to simply substitute the
new planet X values into the formula, and rearrange it:
MX = 6 ⋅ MEarth
rX = 3 ⋅ rEarth
Fgrav,X =
Fgrav,X =
Fgrav,X =
G(6 ⋅ MEarth ) ⋅ m
(3 ⋅ rEarth )2
6 ⋅ G ⋅ MEarth ⋅ m
9 ⋅ (rEarth )2
2 ⎧⎪ G ⋅ MEarth ⋅ m ⎫⎪ 2
⋅⎨
⎬ = ⋅ {550N} = 367N
3 ⎪⎩ (rEarth )2 ⎪⎭ 3
Earth
Planet X
11. A car of mass M coasts down a hill inclined at an angle θ below the horizontal. The car is acted on by three
forces: (i) the normal force N exerted by the road, (ii) a force due to air resistance, Fair, and (iii) the force of gravity,
Mg. Find the total work done on the car as it travels a distance D along the road.
The only forces that can do work on the car are
directed parallel to the incline. The normal force
does no work (EVER).
From the free-body diagram we have two forces
along the hill: the component of gravity downhill is
Mgsinθ and the air resistance points uphill.
D
The work done on the car is the net force times the
distance traveled, so the answer is a)
θ
(Mgsinθ – Fair)D
N
Fair
Mgsinθ
Mgcosθ
Mg
12. When a force of 120.0N is applied to a certain spring, it causes a stretch of 2.25cm. What is the potential
energy of this spring when it is compressed by 3.50cm?
We can use the force formula to find the spring constant: F = kx ⇒ k =
120N
N
= 5333 m
0.0225m
Now use that spring constant in the Potential Energy formula: Uspring = 1 kx 2 =
2
answer c)
1
2
(5333 mN )(0.035m)2 = 3.27J
13. To pass a slow-moving truck on a level road, you want your fancy 1300kg car to accelerate from 13.4m/s to
17.9m/s in 3.00s. What is the minimum power required for this pass?
Power = Work/Time
In this case, the work done on the car will be the change in its kinetic energy.
ΔK = 21 mv 2f − 21 mv i2
ΔK =
1
2
(1300kg)(17.9 ms )2 − 21 (1300kg)(13.4 ms )2 = 91552.5J
Divide by time to get the power.
P=
91552 .5J
= 30517 .5 W
3s
answer c)