HW5: Circular motion and gravity
Due: 1:59am on Thursday, March 22, 2018
You will receive no credit for items you complete after the assignment is due. Grading Policy
PSS 6.1: A Banked Curve
Learning Goal:
To practice Problem-Solving Strategy 6.1: Circular motion
A highway curve with radius R = 274 m is to be banked so that a car traveling v = 25.0 m/s will not skid sideways even in
the absence of friction. At what angle should the curve be banked?
Problem-Solving Strategy 6.1 Circular motion
SET UP
1. Draw a sketch of the physical situation, and identify the moving object or objects to which you will apply
Newton’s second law.
2. Draw a free-body diagram for each chosen object, showing all the forces acting on that object. Label the
magnitudes of unknown forces with algebraic symbols.
3. Show your coordinate axes explicitly in each free-body diagram, and then determine components of forces with
reference to these axes. If you know the direction of the acceleration, it is usually best to take that direction as
one of the axes.
SOLVE
4. Write the equations for Newton’s second law in component form:
ΣFx = max and ΣFy = ma y .
Be careful, when the object is undergoing uniform circular motion. The quantity m (v2 /R) must correspond to
the m a side of ΣF
ma .
=
A serious peril in circular-motion problems is the temptation to regard m ( v2 /R) as a force similar to the forces
exerted by strings, contact with other bodies, or gravitation. Resist this temptation! The quantity m (v2 /R) is
not a real physical force.
5. If more than one object is involved, repeat step 4 for each object.
REFLECT
6. Check particular cases or extreme values of quantities, when possible, and compare the results for these
particular cases with your intuitive expectations. Think about what the problem has taught you that you can
apply to other problems in the future.
SET UP
Before writing any equations, organize your information and draw appropriate diagrams.
Part A
, A sketch of the system is drawn and labeled. Which of the
following is an accurate free-body diagram for the car?
ANSWER:
A
B
C
D
E
Correct
Part B
A sketch of the system depicting all the forces on the car is drawn and labeled. Which coordinate axis (highlighted in red)
is the best choice to use for this problem?
Hint 1. The direction of acceleration
If the object undergoes uniform circular motion, it experiences an acceleration that is directed toward the center
of its circular path.
ANSWER:
, There is no best coordinate axis to use for this problem; either axis is an appropriate choice.
Correct
Part C
What are the components of each force along these axes?
Hint 1. Determining the angle between the normal force and the positive y axis
You are given the angle θ. You now need to determine the angle between the normal force and the positive y axis
in order to find the components of the normal force. To do so, first determine the complement of θ , which is also
the angle between the road and the positive y axis. Then, use this value and the fact that the normal force is
perpendicular to the road to determine the angle between the normal force and the positive y axis.
Due: 1:59am on Thursday, March 22, 2018
You will receive no credit for items you complete after the assignment is due. Grading Policy
PSS 6.1: A Banked Curve
Learning Goal:
To practice Problem-Solving Strategy 6.1: Circular motion
A highway curve with radius R = 274 m is to be banked so that a car traveling v = 25.0 m/s will not skid sideways even in
the absence of friction. At what angle should the curve be banked?
Problem-Solving Strategy 6.1 Circular motion
SET UP
1. Draw a sketch of the physical situation, and identify the moving object or objects to which you will apply
Newton’s second law.
2. Draw a free-body diagram for each chosen object, showing all the forces acting on that object. Label the
magnitudes of unknown forces with algebraic symbols.
3. Show your coordinate axes explicitly in each free-body diagram, and then determine components of forces with
reference to these axes. If you know the direction of the acceleration, it is usually best to take that direction as
one of the axes.
SOLVE
4. Write the equations for Newton’s second law in component form:
ΣFx = max and ΣFy = ma y .
Be careful, when the object is undergoing uniform circular motion. The quantity m (v2 /R) must correspond to
the m a side of ΣF
ma .
=
A serious peril in circular-motion problems is the temptation to regard m ( v2 /R) as a force similar to the forces
exerted by strings, contact with other bodies, or gravitation. Resist this temptation! The quantity m (v2 /R) is
not a real physical force.
5. If more than one object is involved, repeat step 4 for each object.
REFLECT
6. Check particular cases or extreme values of quantities, when possible, and compare the results for these
particular cases with your intuitive expectations. Think about what the problem has taught you that you can
apply to other problems in the future.
SET UP
Before writing any equations, organize your information and draw appropriate diagrams.
Part A
, A sketch of the system is drawn and labeled. Which of the
following is an accurate free-body diagram for the car?
ANSWER:
A
B
C
D
E
Correct
Part B
A sketch of the system depicting all the forces on the car is drawn and labeled. Which coordinate axis (highlighted in red)
is the best choice to use for this problem?
Hint 1. The direction of acceleration
If the object undergoes uniform circular motion, it experiences an acceleration that is directed toward the center
of its circular path.
ANSWER:
, There is no best coordinate axis to use for this problem; either axis is an appropriate choice.
Correct
Part C
What are the components of each force along these axes?
Hint 1. Determining the angle between the normal force and the positive y axis
You are given the angle θ. You now need to determine the angle between the normal force and the positive y axis
in order to find the components of the normal force. To do so, first determine the complement of θ , which is also
the angle between the road and the positive y axis. Then, use this value and the fact that the normal force is
perpendicular to the road to determine the angle between the normal force and the positive y axis.
