A mass m = 6.0 kg is attached to the lower end of a massless string of length L = 73.0 cm. The upper end of the string is held fixed. Suppose that the mass moves in a circle at constant speed, and that the string makes an angle theta = 27o with the vertical, as shown in the figure. How long does it take the mass to make one complete revolution?
see this as an example. https://answers.yahoo.com/question/index?qid=20100218185124AA0oXZ0
Oh, this mass on a string is really twirly! It's like a fancy dance move, but with physics involved. So, we have a mass of 6.0 kg attached to a string of length 73.0 cm. The string makes an angle of 27 degrees with the vertical. We want to find out how long it takes for the mass to make one full revolution.
Well, let's break it down. The angle that the string makes with the vertical is the same as the angle between the string and the horizontal. So, we have a right triangle formed by the string, the vertical line, and the horizontal line. The length of the horizontal line is the radius of the circle that the mass is moving in.
To find the radius, we can use a little trigonometry. The sine of the angle is equal to the opposite side (the vertical line) divided by the hypotenuse (the length of the string). So, we have sin(theta) = opposite/hypotenuse, which gives us sin(27 degrees) = x/73 cm.
Now, we can solve for x (the length of the horizontal line). x = sin(27 degrees) * 73 cm = 31.6 cm.
Since the mass is moving in a circle at a constant speed, we know that the time it takes for one complete revolution is the circumference of the circle divided by the speed of the mass.
The circumference of the circle is 2 * pi * r, where r is the radius of the circle. In this case, r is equal to x which is 31.6 cm. So, the circumference is 2 * pi * 31.6 cm = 198.7 cm.
Now, we just have to convert the circumference to meters (because SI units are cool). 198.7 cm = 1.987 m.
Finally, divide the circumference by the speed of the mass to get the time it takes for one complete revolution. And there you have it, the answer to the question we've been circling around is 1.987 meters divided by the speed of the mass. And I'm sure you already know the speed, right? If not, you might have to do some quick calculations to find it.
Hope this twirling explanation didn't make your head spin too much!
To determine the time it takes for the mass to make one complete revolution, we need to use the relationship between the speed of the mass, the circumference of the circular path it follows, and the time it takes to complete one revolution.
Here's how to find the answer:
1. Start by finding the circumference of the circular path. The circumference of a circle is given by the formula C = 2πr, where r is the radius of the circle. In this case, the radius is equal to the length of the string, L. So the circumference is C = 2πL.
2. Next, we need to find the speed of the mass. Since the mass is moving in a circle at constant speed, we can use the formula v = ωr, where v is the speed, ω (omega) is the angular velocity, and r is the radius of the circle. In this case, the radius is again equal to the length of the string, L. So the speed is v = ωL.
3. Now, we need to relate the angular velocity, ω, to the time it takes to complete one revolution. The angular velocity is defined as the angle covered per unit time. In this case, the mass makes one complete revolution, which corresponds to an angle of 360 degrees or 2π radians. So the angular velocity can be calculated as ω = 2π / t, where t is the time to complete one revolution.
4. Finally, we can combine the equations from steps 2 and 3 to find the time it takes to complete one revolution. Substituting ω = 2π / t and v = ωL, we get v = (2π / t) L. Solving for t, we have t = (2πL) / v.
Plug in the given values for L (73.0 cm) and the angle theta (27 degrees) to find the length of the string corresponding to this angle. Then calculate the speed, v, using the angular velocity formula.
Finally, substitute the values of L and v into the equation t = (2πL) / v to find the time it takes for the mass to make one complete revolution.