Figure P6.57 shows a photo of a swing ride at an amusement park. The structure consists of a horizontal, rotating, circular platform of diameter D from which seats of mass m are suspended at the end of mass less chains of length d. When the system rotates at constant speed, the chains swing outward and make an angle with the vertical. Consider such a ride with the following parameters: D = 8.00 m, d = 2.50 m, m = 10.0 kg , and = 28.0 (a) What is the speed of each seat? (b) Draw a diagram of forces acting on the combination of a seat and a 40.0 - kg child and (c) find tension in the chain.
a) The speed of each seat is 5.19 m/s .
b) The diagram of forces acting on the combination of a seat is shown in step 2.
c) The tension in the chain is 555N.
The expression for the centrifugal force is given by,
Here is the mass of the object, v is the speed and r is the radius, F is the centrifugal force. Centrifugal force is also known by the name of centripetal force.
(a) The below figure shows the direction of forces and direction.
Here F is the centrifugal force, D is the diameter, d is the length of the chain, m is the mass of the seats, M is the mass of the child, T is the tension, is the angle, is the vertical component of the tension and is the horizontal component of the tension.
From the above figure, the radius is equal to
Substitute for , for , for in the above equation.
From the above figure, the expression for the centrifugal force is
Substitute for , for , for m in the above equation.
Now balance the force in horizontal and vertical direction.
Substitute for from equation (1) into the above equation.
The ratio of equation (2) and equation (3) is
Substitute role="math" localid="1663678381997" for , for and for in the above equation.
Therefore, the speed of each seat is .
The total mass is equivalent to the mass of child and seat.
Substitute for and for in the above equation.
The total weight of combination, i.e seat and child is,
Substitute for and .
The horizontal component of the tension is and the vertical component of the tension is .
The below diagram shows the direction of the forces acting on the combination of a seat.
From equation (2).
As, and .
On solving equation
Substitute for and for in the above equation.
Therefore, the tension in the chain is .
Q Question: Initially, the system of objects shown in Figure P5.93 is held motionless. The pulley and all surfaces and wheels are frictionless. Let the force be zero and assume that can move only vertically. At the instant after the system of objects is released, find (a) the tension in the string, (b) the acceleration of , (c) the acceleration of , and (d) the acceleration of . (Note: The pulley accelerates along with the cart.)
The spirit-in-glass thermometer, invented in Florence, Italy, around 1654, consists of a tube of liquid (the spirit) containing a number of submerged glass spheres with slightly different masses (Fig. P14.76). at sufficiently low temperatures, all the spheres float, but as the temperature rises, the spheres sink one after another. The device is a crude but interesting tool for measuring temperature. Suppose the tube is filled with ethyl alcohol, whose density is 0.78945 at and decreases to at . (a) Assuming that one of the spheres has a radius of 1.000 cm and is in equilibrium halfway up the tube at , determine its mass. (b) When the temperature increases to , what mass must the second sphere of the same radius have to be in equilibrium at the halfway point? (c) At, the first sphere has fallen to the bottom of the tube. What upward force does the bottom of the tube exert on this sphere?
84. A thin rod of mass and length is at rest, hanging vertically from a strong, fixed hinge at its top end. Suddenly, a horizontal impulsive force is applied to it. (a) Suppose the force acts at the bottom end of the rod. Find the acceleration of its center of mass and (b) the horizontal force the hinge exerts. (c) Suppose the force acts at the midpoint of the rod. Find the acceleration of this point and (d) the horizontal hinge reaction force. (e) Where can the impulse be applied so that the hinge will exert no horizontal force? This point is called the center of percussion.
A placekicker must kick a football from a point 36.0 m (about 40 yards) from the goal. Half the crowd hopes the ball will clear the crossbar, which is 3.05 m high. When kicked, the ball leaves the ground with a speed of 20.0 m/s at an angle of 53.0° to the horizontal.
(a) By how much does the ball clear or fall short of clearing the crossbar?
(b) Does the ball approach the crossbar while still rising or while falling?
94% of StudySmarter users get better grades.Sign up for free