(a)
To express two vectors in unit vector notation.
(a)
Answer to Problem 67P
The two vectors in unit vector notation is
Explanation of Solution
Write the expression for force from Figure P6.67.
Simplify the above equation.
Write the expression for force from Figure P6.67.
Simplify the above equation.
Conclusion:
Therefore, the two vectors in unit vector notation is
(b)
The total force exerted on the object.
(b)
Answer to Problem 67P
The total force exerted on the object is
Explanation of Solution
Write the total force exerted on the object.
Use equation (I) and (II) in equation (III), to find the net force.
Conclusion:
Therefore, the total force exerted on the object is
(c)
The acceleration of the object.
(c)
Answer to Problem 67P
The acceleration of the object is
Explanation of Solution
Write the expression for acceleration of the object.
Here,
Use equation (IV) in (V).
Given that the mass of the object is
Rearrange the above equation.
Conclusion:
Therefore, the acceleration of the object is
(d)
The velocity of the object.
(d)
Answer to Problem 67P
The velocity of the object is
Explanation of Solution
Write the velocity of the object.
Here,
Conclusion:
Substitute
Therefore, the velocity of the object is
(e)
The position of the object at time
(e)
Answer to Problem 67P
The position of the object at
Explanation of Solution
Given that the initial position of the object is zero.
Write the expression for position using equation of motion.
Here,
Conclusion:
Substitute
Therefore, the position of the object at
(f)
The final kinetic energy of the object.
(f)
Answer to Problem 67P
The final kinetic energy of the object is
Explanation of Solution
Write the expression for kinetic energy.
Here,
From subpart (d) the final velocity vector is given by
Write the magnitude of final velocity vector.
Conclusion:
Substitute
Therefore, the final kinetic energy of the object is
(g)
The final kinetic energy of the object using the equation
(g)
Answer to Problem 67P
The final kinetic energy of the object using the equation
Explanation of Solution
Write the expression for kinetic energy.
From subpart (b) the net force
Apply the above condition in equation (XII).
Conclusion:
Substitute
Therefore, the final kinetic energy of the object using the equation
(h)
To compare the answers in subpart (f) and subpart (g).
(h)
Answer to Problem 67P
The answers in subpart (f) and subpart (g) is same, and it proves that the work energy theorem is consistent with the newton’s second law of motion.
Explanation of Solution
From subpart (f) and (g), the kinetic energy is obtained as
The work energy theorem states that the change in kinetic energy is equal to the external work done, and by rearranging the terms the final kinetic energy of the object is obtained in subpart (g).
Since the answers in subpart (g) and (f) are same, which proves that the work energy theorem is consistent with the newton’s second law of motion.
Conclusion:
Therefore, the answers in subpart (f) and subpart (g) is same, and it proves that the work energy theorem is consistent with the newton’s second law of motion.
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Chapter 6 Solutions
Principles of Physics: A Calculus-Based Text
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