Concept explainers
The 2-lb ball at A is suspended by an inextensible cord and given an initial horizontal velocity of v0. If l = 2 ft, xB = 0.3 ft, and yB = 0.4 ft, determine the initial velocity v0 so that the ball will enter the basket. (Hint: Use a computer to solve the resulting set of equations.)
Fig. P13.201
Find the initial velocity
Answer to Problem 13.201RP
The initial velocity
Explanation of Solution
Given information:
The weight of the ball (m) is
The length of the cord (l) is
The horizontal distance between the basket and point of suspension of the ball
The vertical distance between basket and point of suspension of the ball
The acceleration due to gravity (g) is
Calculation:
Show the diagram of the suspended ball by an inextensible cord as in Figure (1).
Assume that position ‘1’ be at A and position ‘2’ be at the point described by the angle where the path of the ball changes from circular to parabolic.
The tension in the cord at position ‘2’, becomes slack
Refer Figure (1),
The expression for the x-coordinate of the ball at position ‘2’
The expression for the y-coordinate of the ball at position ‘2’
Show the free body diagram of the ball at position ‘2’ as in Figure (2).
The expression for the normal acceleration of the ball
Since, the cord becomes slack at position ‘2’, so the tension (Q) will be zero.
Calculate the velocity of the ball by applying Newton’s second law and resolve the forces acting on the ball at position ‘2’ using the relation:
Substitute
Substitute 0 for
The expression for the kinetic energy of the ball at position ‘1’
Calculate the potential energy of the ball at position ‘1’
Here, negative sign is used as the ball is located below the datum level and h is the vertical distance of the ball from the datum level.
Substitute
The expression for the kinetic energy of the ball at position ‘2’
The expression for the vertical distance of the ball by referring the Figure 1 as follows:
Calculate the potential energy of the ball at position ‘2’
Substitute
The expression for principle of conservation of energy at position ‘1’ and position ‘2’ for the ball, to calculate the angle swept by the ball
Substitute
Find the velocity at position 2:
Substitute
Show the parabolic motion of the ball after it reaches position ‘2’ as in Figure (3).
The expression for the velocity of the projectile ball after reaching the position ‘2’
Here,
The expression for the horizontal velocity component of the projectile ball along the negative X-axis as follows:
The expression for the horizontal distance between the basket and point of suspension of the ball
Here,
Substitute
Substitute
The expression for the vertical velocity component of the projectile ball along the negative X-axis as follows:
The expression for the vertical distance between the basket and point of suspension of the ball
Substitute
Substitute
Use trial and error method to calculate the value of
Case (1):
Try
Find the velocity at position 2:
Substitute
Find the time difference between basket and ball:
Substitute
Find the vertical distance between basket and ball:
Substitute
Case (2):
Try
Find the velocity at position 2:
Substitute
Find the time difference between basket and ball:
Substitute
Find the vertical distance between basket and ball:
Substitute
Case (3):
Try
Find the velocity at position 2:
Substitute
Find the time difference between basket and ball:
Substitute
Find the vertical distance between basket and ball:
Substitute
The expression for the data point as follows:
From above calculations, the following set of data points is obtained.
Calculate the general form of quadratic Equation.
Here, a, b, c are constants.
Substitute 0 for
Substitute
Substitute
Solve the equation (7) and equation (8).
Substitute
Substitute
Solve the above equation.
Calculate the angle
Substitute
Find the velocity at position 2:
Substitute
Find the initial velocity
Substitute
Therefore, the initial velocity
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Chapter 13 Solutions
Connect 2 Semester Access Card for Vector Mechanics for Engineers: Statics and Dynamics
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