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
Want to see more full solutions like this?
Chapter 13 Solutions
VEC MECH 180-DAT EBOOK ACCESS(STAT+DYNA)
- An electric motor M is used to reel in cable and hoist a bicycle into the ceiling space of a garage. Pulleys are fastened to the bicycle frame with hooks at locations A and B, and the motor can reel in cable at a steady rate of 20 in./sec. At this rate, how long will it take to hoist the bicycle 7.9 feet into the air? Assume that the bicycle remains level.arrow_forwardQuestion 7 [3/155]: The collar of mass m slides on the rough horizontal shaft under the action of the force F of constant magnitude Farrow_forwardPart A The 12-lb box slides on the surface for which u = 0.3. The box has a velocity v = 15 ft /s when it is 2 ft from the plate. (Figure 1) If the box strikes the smooth plate, which has a weight of 19 lb and is held in position by an unstretched spring of stiffness k = 330 lb / ft, determine the maximum compression imparted to the spring. Take e = 0.8 between the box and the plate. Assume that the plate slides smoothly. Express your answer using three significant figures and include the appropriate units. HA ? Value Units Submit Request Answer Figure < 1 of 1 < Return to Assignment Provide Feedback v = 15 ft/s 2 ftarrow_forwardA team of engineering students is designing a catapult to launch a small ball at A so that it lands in the box. If it is known that the initial velocity vector makes a 0 = 38° angle with the horizontal, determine the range of launch speeds for which the ball will land inside the box. Assume a = 10.4 ft, b = 2 ft, c = 5 in., d = 12 in. Answers:( i +6+ ≤00 ≤ i ) ft/secarrow_forwardA 15 -kg collar travels down a smooth slope which is described by the function: y = 0.1 x2 as shown. If at location 1 its speed is 15 m/s, determine its speed when it gets to position 2, The spring has original length (unstretched length) = 7 m and stiffness k=60 N/m. Note: you must use Principle of Work and Energy y = 0.1 x? 1 2 6 т + 10 marrow_forwardA truck with mass of M is going to tow the car with mass m. For values of M and m please follow the Table 1. If the car is initially at rest and the truck is moving with a velocity of 30 km/h when the cable is still slack, (i) Draw the free-body diagram of the truck and car system. (ii) (iii) determine the common velocity of the truck and the car just after the cable becomes taut. (note: the velocity of truck and car would be the same). find the loss of energy during this process. km h 30- First name starting (A-M) First name starting (N-Z) Surname starting (A-M) Surname starting (N-Z) Figure 1 Table 1 * M = 1,500 kg M = 2,500 kg m = 1,000 kg m = 800 kgarrow_forwardA 7-lb collar is attached to a spring and slides without friction along a rod in the vertical plane. The spring has a constant 3.5 lb/in and 15-in. undeformed length. The dimensions are: h = 15 in. The collar is moving to the left at v = 2 in/s in the position shown. Determine the force exerted by the rod on the collar at (a) point A, (b) point B. Both A and B are on the curved portion of the rod. h www. LA h B h harrow_forwardA motorcycle and rider have a combined mass of 350kg. The vehicle's velocity is 100 km/hr. The rider is to go up a hill incline of 10 meters. The wheels each have a mass of 20 kg and a diameter of 500mm. The wheel design consists of 6 spokes with a mass of 0.5 kg for each spoke. a. Determine the vehicle's velocity at the top of the hill, assuming it was a rolling vehicle with engine being turned off. a. Indicate the vehicle's velocity at each meter through the climb. b. Explain the transfer energy and how this affects the behaviour of the system. At another point in its journey, the motorcycle and rider travel at 80 km/h around a left-hand bend of radius 30m. Calculate: a) The angular velocity of each wheel. b) The moment of inertia of each wheel. c) The angular momentum of the wheel prior to entering the bend. d) The magnitude of the gyroscopic torque produced on the bike as the rider is driving around the bend. What is the effect and why is it important to calculate the gyroscopic…arrow_forwardThe two spheres A and B each have a mass of 400 g. The spheres are fixed to the horizontal rods as shown in (Figure 1) and their initial velocity is 2 m/s. The mass of the supporting frame is negligible and it is free to rotate. Neglect the size of the spheres. Part A: If a couple of moment of M = 0.7 N⋅m is applied to the frame, determine the speed of the spheres in 3 s. Express your answer to three significant figures and include the appropriate units.arrow_forwardSlider C has a mass of 0.5 kg and may move in a slot cut in arm AB, which rotates at constant speed in a horizontal plane. The slider is attached to a spring of constant k = 150 N/m, which is unstretched when r = 0. When arm AB rotates about the vertical axis, the slider moves without friction outward along the smooth slot cut. Determine for the position r = 80 mm: a) The constant speed (V) of the slider. b) The normal force (N) exerted on the slider by arm AB. A r=80mm Barrow_forwardThe "flying car" is a ride at an amusement park which consists of a car having wheels that roll along a track mounted inside a rotating drum. By design the car cannot fall off the track, however motion of the car is developed by applying the car's brake, thereby gripping the car to the track and allowing it to move with a constant speed of the track, vt = 3 m/s. The rider applies the brake when going from B to A and then releases it at the top of the drum, A, so that the car coasts freely down along the track to B (0 = π rad). Neglect friction during the motion from A to B. The rider and car have a total mass of 390 kg and the center of mass of the car and rider moves along a circular path having a radius of R = 9.8 m. (Figure 1) Figure R Barrow_forwardQ6: The 7-kg collar A slides with negligible friction Q8: The 1.5-kg ball is given an initial velocity va = 2.5m/s in the vertical plane at position A, where the two horizontal attached springs are unstretched. The ball follows the dashed path shown and crosses point B, which is 125 mm. directly below A. Calculate the velocity vg of the ball at B. Each spring has a stiffness of 1800 N/m. on the fixed vertical shaft. When the collar is released from rest at the bottom position shown, it moves up the shaft under the action of the constant force F = 200N applied to the cable. Calculate the stiffness k which the spring must have if its maximum compression is to be limited to 75 mm. The position of the small pulley at B is fixed. [ Ans.: k = 8.79 kN/m] [ Ans.: vg = 8.54 ft/s] 225 mm 300mm 300mm 125mm 450 mm 75 mm Q7: Determine the constant force P required to cause the 0.5-kg slider to have a speed v, = 0.8 m/s at position 2. The slider starts from rest at position 1 and the spring has…arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
- Elements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University PressMechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSONThermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill Education
- Control Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEYMechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage LearningEngineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEY