A 160-lb gymnast is executing a series of full-circle swings on the horizontal bar. In the posit

A 160-lb gymnast is executing a series of full-circle swings...

Key Concepts

Conservation of Energy

In rotational dynamics, conservation of energy relates the potential energy lost by a body as it moves under gravity to the gain in its kinetic energy. For a swinging gymnast, this means that as he descends from a higher to a lower position, the decrease in gravitational potential energy is converted into rotational kinetic energy, which can then be used to find his angular speed at different positions in the swing.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its angular velocity and depends on how its mass is distributed with respect to the axis of rotation. In the context of a gymnast performing a swing, the body can be approximated as a rigid body with a certain moment of inertia, which is crucial for analyzing the rotational motion and computing the angular kinetic energy as a function of angular velocity.

Radius of Gyration

The radius of gyration is a derived quantity that represents the distance from the axis of rotation at which the entire mass of the body could be assumed to be concentrated to yield the same moment of inertia. It simplifies the calculation of the moment of inertia in complex bodies by reducing the distributed mass to a single point, making it easier to analyze rotational dynamics in problems like the swinging gymnast.

Angular Velocity

Angular velocity is a measure of how fast an object rotates or revolves relative to another point, typically the center of rotation. In problems involving swinging motions such as that of a gymnast, the angular velocity is determined by how the conversion of gravitational potential energy into rotational kinetic energy increases the speed of rotation, and it is directly related to the forces exerted during the motion.

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path and is directed towards the center of the circle. For a gymnast on a swinging bar, this force is provided by the hands and internal forces, and its magnitude depends on the angular velocity and the radial distance from the pivot point. The centripetal force is critical for maintaining the circular motion and is the sum of the forces, including body weight and inertial effects from the rotation.

Transcript

00:01 Hello everyone as shown in the figure at jamnast of page 160 power is executing series of full -circle swing in the horizontal bar in the position zone here small and negligible clockwise angular velocity so angular velocity is negligible and maintain body straight and rigid as he swings downward assuming that during the swing the central radial radius of gyration of his body is 1 .5th.

00:48 Calculate angular velocity and force accepted on his hand.

00:56 We have to calculate angular velocity and force exerted on the hand as he rotate through.

01:24 First part, 90 degrees.

01:29 Second part, 180 degree.

01:36 Position 1.

01:37 We are calling this is to be one position and this is the second position position one directly above the bar at 1 3 .5 fifth so gravitational potential energy weight into height of center of gravity that is 160 into 3 .5 so it is 565 pounds omega 1 is 0 and v1 is also 0 so kinetic energy t1 will be also 0 first part position 2 that is horizontal position itra becomes nitrogen negative h2 is 0 gravitational potential energy is 0 b2 will be 3 point types of omega 2 so kinetic energy will be half m b2 square half i omega square that is m k square omega 2 square substitute the value mass of the man is 160 pound is the way converting into mass 3 .5 omega 2 square mass 160 upon 32 .2 k k is given 1 .5 so this total kinetic energy at position b, we will get a2 is equal to 36 .025 omega 2 squared.

05:03 Applying conservation of energy, t1 plus b1 is called to t2 plus b2.

05:30 T1 initial kinetic energy in vertical position is 0.

05:35 B1 is 560.

05:38 B2 is 36 .025 omega 2 squared, v2 is 0.

05:51 So omega 2 from this above equation you will get 3 .94 radiant per second in clock by the now we have to calculate the force on the arms with a free burn diagram.

06:26 Here reaction on the arms is ry and rx.

06:33 It's the way mg will act here, i into alpha as m8 normal.

07:06 Tendential acceleration will be 3 .5 alpha.

07:12 Normal acceleration is 3 .5 omega 2 square, 54 .407, 5 into second square towards left.

07:51 Now calculating moment about o taking clock by direction to be positive 3 .5 into 160 to be m omega m tangential, that is 160.

09:12 So value of alpha you will get 7 .7724, a gradient per second square.

09:30 And tangential acceleration is 7 .7724.

09:44 Into a .5 that is 27 .203 fifth per second square in downward direction...