Consider the conical pendulum, a mass on the end of a string, with the other end of the string fixed to the ceiling. Given the proper push, this pendulum can swing in a circle at an angle q of 25.5 with respect to the vertical, maintaining the same height throughout its motion. If the mass of the pendulum M is 14.6 kg and the length of the string L is 0.5 m, what is the speed of the mass as it swings?

Answer 1

We can solve this problem using centripetal force and Newton's laws of motion. The centripetal force acting on the mass is given by the formula:

F_c = M * v^2 / r

where M is the mass of the pendulum, v is its speed, and r is the radius of the circle that the mass traces out.

To find the radius, we can use trigonometry. The radius is equal to the length of the string multiplied by the sine of the angle:

r = L * sin(q) = 0.5m * sin(25.5) = 0.214m

Now, we can look at the forces acting on the mass in the vertical direction. There are two forces acting on the mass: the tension force (T) upward and the gravitational force (M * g) downward. Since the mass remains at a constant height, these forces must balance each other:

T * cos(q) = M * g

where g is the acceleration due to gravity (approximately 9.81 m/s²).

Now we can look at the horizontal forces. The only horizontal force is the component of tension force, which provides the centripetal force:

T * sin(q) = M * v^2 / r

We can solve for tension (T) using the vertical force equation and plug it into the horizontal force equation:

(M * g) / cos(q) * sin(q) = M * v^2 / r

Now we can solve for v:

v^2 = (r * g * tan(q)) / cos(q)

v^2 = (0.214 * 9.81 * tan(25.5)) / cos(25.5)

v^2 = 6.9629

v = sqrt(6.9629)

v ≈ 2.64 m/s

So, the speed of the mass as it swings is approximately 2.64 m/s.