A ball of mass m = 0.2 kg is attached to a (massless) string of length L = 2 m and is undergoing circular motion in the horizontal plane (imagine a wave swinger ride at the amusement park).
What should the speed of the mass be for θ to be 39°? What is the tension in the string?
The answers are 3.16 m/s and 2.52 N. How do I solve for these answers?
It all follows from Newton's second law F = m a. The mass is rotating in a cricle with radius r = L sin(theta), so it's acceleration is:
a = v^2/r = v^2/[L sin(theta)]
which has a directon toward the center of the circle.
This means that a total force of
F = m a = m v^2/[L sin(theta)]
is acting on the mass, which points toward the center.
Then the source of this force is the string tension. If the string tension is T, then the horizontal component of this force is T sin(theta) and this must be equal to F:
T sin(theta) = m v^2/[L sin(theta)]
The total force in the vertical direction is zero, as the mass is not accelerating in that direction. Now, the string tension does have a comnponent in that direction of
T cos(theta). The gravitational force also acts in the mass, it acts with a foce of m g, directed downward. So, the total force in the vertical direction is
T cos(theta) - m g and thus has to be zero:
T cos(theta) - m g = 0 --->
T cos(theta) = m g
If you divide the two equations, you get:
T sin(theta)/[T cos(theta)] =
m v^2/[L sin(theta)] / (m g) --->
v = sin(theta)sqrt[L g /cos(theta)]
= 3.16 m/s
The string tension can be computed from e.g. T cos(theta) = m g, this gives T = 2.52 N
Thank you.
To solve for the speed of the mass and the tension in the string, we can apply the principles of circular motion.
1. Speed of the mass:
The speed of the mass can be determined using the concept of centripetal acceleration. The centripetal acceleration is given by the equation:
a = v^2 / r,
where "a" is the centripetal acceleration, "v" is the speed of the mass, and "r" is the radius of the circular path.
In this case, the radius is equal to the length of the string, L. So, r = L = 2 m. The acceleration can be determined by using the vertical component of the gravitational force acting on the mass, which is given by:
mg * sinθ,
where "m" is the mass of the ball, and "g" is the acceleration due to gravity. Since the ball is undergoing circular motion in the horizontal plane, only the vertical component of the gravitational force contributes to the centripetal acceleration.
Setting these two equations equal to each other, we get:
mg * sinθ = v^2 / r.
Plugging in the given values, m = 0.2 kg and θ = 39°, we have:
0.2 * 9.8 * sin39° = v^2 / 2.
Simplifying the equation, we can rearrange it to solve for v:
v^2 = (0.2 * 9.8 * sin39°) * 2.
Taking the square root of both sides, we find:
v = √((0.2 * 9.8 * sin39°) * 2).
Evaluating this expression, we find:
v ≈ 3.16 m/s.
Therefore, the speed of the mass should be approximately 3.16 m/s.
2. Tension in the string:
The tension in the string can be determined using the centripetal force, which is provided by the tension in the string itself. The centripetal force is given by the equation:
Fc = m * a,
where "Fc" is the centripetal force, "m" is the mass of the ball, and "a" is the centripetal acceleration.
Substituting the known values, Fc = 0.2 kg * (0.2 * 9.8 * sin39°).
Simplifying this equation, we find:
Fc ≈ 0.2 kg * 1.961 * 0.6293.
Evaluating the expression, we get:
Fc ≈ 0.252 N.
Therefore, the tension in the string is approximately 0.252 N.