2) Suppose you put an object on an inclined slope and let go. If there is enough friction between the object and the surface of the slope, it might not go anywhere. This can be important when figuring out what kind of tires you need so you don't slip on an icy hill! Whether there is friction or not, it is useful to be able to write the gravitational force acting on the object as the sum of two perpendicular vectors: one parallel to the surface of the slope, and one perpendicular to the surface of the slope. For this series of questions, suppose the inclined slope is fixed at an angle of 30° and the object has a mass of 3 kg, as shown in the diagram. 30° 3 kg a) Gravitational force has a magnitude equal to an object's mass times acceleration due to gravity, and a direction of "straight down." Since we are using kilograms, acceleration due to gravity should be given in meters per second squared, and the value is approximately 9.8 m/s². Use this information to write the components of vector F, the force of gravity acting on the object. Remember that "down" is usually associated with a negative direction. b) Use vector projection to find the part of vector F that is parallel to the inclined surface of the slope. Call this vector ū. (Hint: you may find it useful to first write any vector in the direction of the slope, call it vector v, and use that in projection calculations.) c) Use your answers to (a) and (c) to write a vector w which is perpendicular to the surface of the slope (that is, perpendicular to your vector v) and which, when added to ū, results in the gravitational force vector F. In other words, u + w should equal F. d) Draw a picture showing the slope, the object, the gravitational force F, and the vectors ū and w.
2) Suppose you put an object on an inclined slope and let go. If there is enough friction between the object and the surface of the slope, it might not go anywhere. This can be important when figuring out what kind of tires you need so you don't slip on an icy hill! Whether there is friction or not, it is useful to be able to write the gravitational force acting on the object as the sum of two perpendicular vectors: one parallel to the surface of the slope, and one perpendicular to the surface of the slope. For this series of questions, suppose the inclined slope is fixed at an angle of 30° and the object has a mass of 3 kg, as shown in the diagram. 30° 3 kg a) Gravitational force has a magnitude equal to an object's mass times acceleration due to gravity, and a direction of "straight down." Since we are using kilograms, acceleration due to gravity should be given in meters per second squared, and the value is approximately 9.8 m/s². Use this information to write the components of vector F, the force of gravity acting on the object. Remember that "down" is usually associated with a negative direction. b) Use vector projection to find the part of vector F that is parallel to the inclined surface of the slope. Call this vector ū. (Hint: you may find it useful to first write any vector in the direction of the slope, call it vector v, and use that in projection calculations.) c) Use your answers to (a) and (c) to write a vector w which is perpendicular to the surface of the slope (that is, perpendicular to your vector v) and which, when added to ū, results in the gravitational force vector F. In other words, u + w should equal F. d) Draw a picture showing the slope, the object, the gravitational force F, and the vectors ū and w.
Classical Dynamics of Particles and Systems
5th Edition
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Stephen T. Thornton, Jerry B. Marion
Chapter10: Motion In A Noninertial Reference Frame
Section: Chapter Questions
Problem 10.13P
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Transcribed Image Text:2) Suppose you put an object on an inclined slope and let go. If there is enough friction between the
object and the surface of the slope, it might not go anywhere. This can be important when figuring out
what kind of tires you need so you don't slip on an icy hill!
Whether there is friction or not, it is useful to be able to write the
gravitational force acting on the object as the sum of two perpendicular
vectors: one parallel to the surface of the slope, and one perpendicular
to the surface of the slope.
For this series of questions, suppose the inclined slope is fixed at an
angle of 30° and the object has a mass of 3 kg, as shown in the diagram.
30°
3 kg.
a)
Gravitational force has a magnitude equal to an object's mass times acceleration due to gravity,
and a direction of "straight down." Since we are using kilograms, acceleration due to gravity
should be given in meters per second squared, and the value is approximately 9.8 m/s². Use this
information to write the components of vector F, the force of gravity acting on the object.
Remember that "down" is usually associated with a negative direction.
b) Use vector projection to find the part of vector F that is parallel to the inclined surface of the
slope. Call this vector u. (Hint: you may find it useful to first write any vector in the direction of
the slope, call it vector , and use that in projection calculations.)
c) Use your answers to (a) and (c) to write a vector w which is perpendicular to the surface of the
slope (that is, perpendicular to your vector v) and which, when added to ū, results in the
gravitational force vector F. In other words, û + w should equal F.
d) Draw a picture showing the slope, the object, the gravitational force F, and the vectors ū and ŵ.
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