Category Archives: Engineering Mathematics Blog
Problem 6-8: An integrated problem involving circular motion, momentum, and projectile motion
Integrated Concepts
When kicking a football, the kicker rotates his leg about the hip joint.
(a) If the velocity of the tip of the kicker’s shoe is 35.0 m/s and the hip joint is 1.05 m from the tip of the shoe, what is the shoe tip’s angular velocity?
(b) The shoe is in contact with the initially stationary 0.500 kg football for 20.0 ms. What average force is exerted on the football to give it a velocity of 20.0 m/s?
(c) Find the maximum range of the football, neglecting air resistance.
Solution:
Part A
From the given problem, we are given the following values: and . We are required to solve for the angular velocity .
The linear velocity, and the angular velocity, are related by the equation
If we substitute the given values into the formula, we can directly solve for the value of the angular velocity. That is,
Part B
For this part of the problem, we are going to use Newton’s second law of motion in term of linear momentum which states that the net external force equals the change in momentum of a system divided by the time over which it changes. That is
For this problem, we are given the following values: , , , and . Substituting all these values into the equation, we can solve directly for the value of the net external force.
Part C
This is a problem on projectile motion. In this particular case, we are solving for the range of the projectile. The formula for the range of a projectile is
We are asked to solve for the maximum range, and we know that the maximum range happens when the angle is .
Problem 6-7: Calculating the angular velocity of a truck’s rotating tires
A truck with 0.420-m-radius tires travels at 32.0 m/s. What is the angular velocity of the rotating tires in radians per second? What is this in rev/min?
Solution:
The linear velocity, and the angular velocity are related by the equation
From the given problem, we are given the following values: and . Substituting these values into the formula, we can directly solve for the angular velocity.
Then, we can convert this into units of revolutions per minute:
Problem 6-6: Calculating the linear velocity of the lacrosse ball with the given angular velocity
In lacrosse, a ball is thrown from a net on the end of a stick by rotating the stick and forearm about the elbow. If the angular velocity of the ball about the elbow joint is 30.0 rad/s and the ball is 1.30 m from the elbow joint, what is the velocity of the ball?
Solution:
The linear velocity, and the angular velocity, of a rotating object are related by the equation
From the given problem, we have the following values: and . Substituting these values in the formula, we can directly solve for the linear velocity.
Problem 6-5: Calculating the angular velocity of a baseball pitcher’s forearm during a pitch
A baseball pitcher brings his arm forward during a pitch, rotating the forearm about the elbow. If the velocity of the ball in the pitcher’s hand is 35.0 m/s and the ball is 0.300 m from the elbow joint, what is the angular velocity of the forearm?
Solution:
We are given the linear velocity of the ball in the pitcher’s hand, , and the radius of the curvature, . Linear velocity and angular velocity are related by
If we substitute the given values into our formula, we can solve for the angular velocity directly. That is,
The angular velocity of the forearm is about 117 radians per second.
Problem 6-4: Period, angular velocity, and linear velocity of the Earth
(a) What is the period of rotation of Earth in seconds? (b) What is the angular velocity of Earth? (c) Given that Earth has a radius of 6.4×106 m at its equator, what is the linear velocity at Earth’s surface?
Solution:
Part A
The period of a rotating body is the time it takes for 1 full revolution. The Earth rotates about its axis, and complete 1 full revolution in 24 hours. Therefore, the period is
Part B
The angular velocity is the rate of change of an angle,
where a rotation takes place in a time .
From the given problem, we are given the following: , and . Therefore, the angular velocity is
We can also express the angular velocity in units of radians per second. That is
Part C
The linear velocity , and the angular velocity are related by the formula
From the given problem, we are given the following values: , and . Therefore, the linear velocity at the surface of the earth is
Problem 6-3: Calculating the number of revolutions given the tires radius and distance traveled
An automobile with 0.260 m radius tires travels 80,000 km before wearing them out. How many revolutions do the tires make, neglecting any backing up and any change in radius due to wear?
Solution:
The rotation angle is defined as the ratio of the arc length to the radius of curvature:
where arc length is distance traveled along a circular path and is the radius of curvature of the circular path.
From the given problem, we are given the following quantities: , and .
Problem 6-2: Conversion of units from rpm to revolutions per second and radians per second
Microwave ovens rotate at a rate of about 6 rev/min. What is this in revolutions per second? What is the angular velocity in radians per second?
Solution:
This is a problem on conversion of units. We are given a rotation in revolutions per minute and asked to convert this to revolutions per second and radians per second.
For the first part, we are asked to convert 6 rev/min to revolutions per second.
For the next part, we are going to convert 6 rev/min to radians per second.

Chapter 8: Partial Differentiation
8.1 Partial Derivative
8.2 Geometric Interpretation of Partial Derivative
Exercise 8.1
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
8.3 Partial Derivatives of Higher Order
Exercise 8.2
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
8.4 Total Differentiation
Exercise 8.3
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
8.5 Total Derivative
Exercise 8.4
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
8.6 Partial Differentiation of Implicit Functions
Exercise 8.5
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
8.7 Tangent Plane and Normal Line
Exercise 8.6
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
8.8 Maxima and Minima
Exercise 8.7
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13

Chapter 7: Derivatives from Parametric Equations, Radius and Center of Curvature
7.1 Derivatives in Parametric Form
Exercise 7.1
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
7.2 Differential of Arc Length
7.3 Radius of Curvature
Exercise 7.2
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
7.4 Center of Curvature
Exercise 7.2
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
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