A 0.25kg stone tied to a rope is swung to make a circular motion in a vertical plane. If the circle is of radius 300cm, what is the minimum uniform speed of the stone that will keep the rope taut.
To determine the minimum uniform speed required to keep the rope taut, we need to analyze the forces acting on the stone.
1. First, we need to determine the tension in the rope. At the minimum speed, the tension in the rope will be at its maximum value.
2. The tension in the rope is responsible for providing the necessary centripetal force, which keeps the stone moving in a circular path. Therefore, we can equate the centripetal force to the tension in the rope.
Centripetal force (F) = Tension in the rope
3. The centripetal force is given by the formula:
F = (mass x velocity^2) / radius
4. Since we want to find the minimum uniform speed, we assume the minimum radius, which is the radius of the circular path the stone is swinging in.
Given:
- Mass of the stone (m) = 0.25 kg
- Radius (r) = 300 cm = 3 m
We can now solve for the minimum uniform speed (v).
Tension (T) = F
F = (m x v^2) / r
T = (m x v^2) / r
T = (0.25 kg x v^2) / 3 m
Now, we have the equation for the tension in terms of the speed of the stone.
To keep the rope taut, the tension must be non-zero. Therefore, the minimum uniform speed of the stone (v) can be found by setting the tension (T) to a value greater than zero.
Let's assume a very small tension value, such as 0.01 N.
0.01 N = (0.25 kg x v^2) / 3 m
Cross multiplying and rearranging the equation, we get:
0.01 N x 3 m = 0.25 kg x v^2
0.03 Nm = 0.25 kg x v^2
v^2 = (0.03 Nm) / 0.25 kg
v^2 = 0.12 Nm/kg
Now, we can take the square root of both sides to solve for v:
v = sqrt(0.12 Nm/kg)
v ≈ 0.346 m/s
Therefore, the minimum uniform speed of the stone that will keep the rope taut is approximately 0.346 m/s.
To find the minimum uniform speed of the stone that will keep the rope taut, we first need to understand the concept of tension in a circular motion.
When an object is in circular motion, there is a centripetal force acting on it, which is directed towards the center of the circle. In this case, the tension in the rope provides the necessary centripetal force to keep the stone moving in a circular path.
The centripetal force (Fc) can be calculated using the equation:
Fc = (m * v^2) / r,
where m is the mass of the stone, v is its velocity, and r is the radius of the circle.
In this problem, m = 0.25 kg and r = 300 cm = 3 meters.
We need to find the minimum velocity (v) that will keep the rope taut, which means the tension in the rope is not zero.
At the minimum, the tension in the rope is equal to zero. So, we can set the tension equal to zero and solve for velocity:
0 = (m * v^2) / r.
Rearranging the equation, we obtain:
v^2 = (r * 0) / m,
v^2 = 0,
v = 0.
This means that the minimum uniform speed of the stone needed to keep the rope taut is zero. Thus, the stone can be at rest, and the rope will remain taut.