A gymnist is swinging on a high bar. The distance between his waist and the bar is 1.1M. At the top of the swing his speed is momentarily zero. ignoring friction and treating the gymnist as if all his mass is located at his waist, find his speed at the bottom of the swing.
At the bottom of his swing, he has lost gravitational potential energy
PE = 2 M g * 1.1 m
If all his mass were at his waist, the velocity of the waist would be given by
KE = (1/2)M V^2 = 2 M g * 1.1 m,
because the potential energy loss would be cnverted to kinetic energy.
Cancel out the unknown M and solve for V.
V = sqrt (4*g*1.1 m)
Well, it seems like the gymnast has swung so high that there's no denying that gravity is just pulling them down with a vengeance. And without any friction or concerns about the gymnast's weight distribution, it's all about that potential energy turning into some good old kinetic energy.
But let's not forget our good friend, Mr. Energy Conservation. We can equate the gravitational potential energy loss to the kinetic energy gain. Mathematically speaking, it's like saying "Hey, potential energy, you're gonna become kinetic energy, deal with it!"
Now, let's get to the point, the speed at the bottom of the swing. We know that the potential energy loss is given by:
PE = 2mg(1.1m)
And that's gonna be equal to the kinetic energy at the bottom, which we can write as:
KE = (1/2)mv^2
Now, let the cancelation of the unknown mass superhero come to rescue us! Since the mass magically vanishes, we can say "Sayonara, M!" and solve for V.
V = √(4g(1.1m))
Voila! There's your speed at the bottom of the swing, my friend. It's the square root of 4 times gravity times the distance from the waist to the bar. Just make sure not to swing too high, or you might end up with a whole new definition of "flying without wings."
To find the speed at the bottom of the swing, we can use the conservation of mechanical energy. At the top of the swing, the gymnast's speed is momentarily zero, so all the energy is in the form of gravitational potential energy. At the bottom of the swing, all the potential energy is converted to kinetic energy.
The gravitational potential energy lost by the gymnast at the bottom of the swing is given by:
PE = mgh = 2mg(1.1m)
Where m is the mass of the gymnast, g is the acceleration due to gravity, and h is the distance between the waist and the bar.
The kinetic energy at the bottom of the swing is given by:
KE = (1/2)mv^2
Where v is the speed at the bottom of the swing.
Since the potential energy loss is converted to kinetic energy, we have:
PE = KE
2mg(1.1m) = (1/2)mv^2
Canceling out the mass m, we can solve for v:
2g(1.1m) = (1/2)v^2
Now we can solve for v:
v^2 = 4g(1.1m)
Taking the square root of both sides, we get:
v = √(4g(1.1m))
Therefore, the speed at the bottom of the swing is √(4g(1.1m)), where g is the acceleration due to gravity (approximately 9.8 m/s^2).
To find the speed of the gymnast at the bottom of the swing, we can start by equating the gravitational potential energy he loses at the top of the swing to the kinetic energy he gains at the bottom of the swing.
The potential energy lost can be calculated using the formula: PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height or distance between the waist and the bar.
In this case, the height is given as 1.1 m, so the potential energy lost is: PE = mg * 1.1 m.
Next, we equate this potential energy loss to the kinetic energy gained at the bottom of the swing. The kinetic energy can be calculated using the formula: KE = (1/2)mv^2, where m is the mass (which we can assume is located at the waist), and v is the velocity or speed.
By setting the potential energy loss equal to the kinetic energy gained, we have: PE = KE.
Substituting the values we know, we have: mg * 1.1 m = (1/2)mv^2.
Now, we can cancel out the unknown mass (M) from both sides of the equation.
This gives us: g * 1.1 m = (1/2)v^2.
Finally, to solve for the speed (v), we can rearrange the equation and take the square root of both sides.
The speed (v) at the bottom of the swing is given by: v = √(2g * 1.1 m).
Simply calculate this expression to find the speed.