To determine the minimum speed required to just cross the topmost position, we need to consider the forces acting on the stone at this point. At the topmost position, the gravitational force is acting downward, while the tension in the string is acting upward.
To keep the stone moving in a circle, the tension in the string must be greater than the gravitational force. At the topmost position, this means that the tension must be equal to the gravitational force plus some extra force to provide the necessary centripetal acceleration.
The gravitational force on the stone is given by:
F_gravity = m * g
where m is the mass of the stone and g is the acceleration due to gravity.
The tension in the string is given by:
F_tension = m * (v^2 / r)
where v is the speed of the stone, and r is the radius of the circle (which is equal to the length of the string).
Setting these two forces equal, we have:
m * g = m * (v^2 / r) + m * g_extra
where g_extra is the extra force needed to provide the necessary centripetal acceleration.
Simplifying, we get:
v^2 = g * r + g_extra
To just cross the topmost position, the centripetal acceleration must be equal to the gravitational acceleration, so we have:
g_extra = g
Substituting this into the previous equation, we get:
v^2 = 2 * g * r
Taking the square root of both sides, we get:
v = sqrt(2 * g * r)
Substituting r = 5 m and g = 9.81 m/s^2, we get:
v = sqrt(2 * 9.81 m/s^2 * 5 m) = 9.9 m/s
Therefore, the minimum speed required to just cross the topmost position is approximately 9.9 m/s.
A stone is tied at one end of a 5 m long string and whirled in a vertical circle. The minimum speed required to just cross the topmost position is
5 answers
the point is for ZERO net string force at the top
weight = mg down = mv^2/r down
There is ZERO tension in the string at that top point
v^2 / r = g
I agree with the numbers, not the logic.
weight = mg down = mv^2/r down
There is ZERO tension in the string at that top point
v^2 / r = g
I agree with the numbers, not the logic.
You are correct, my mistake. At the topmost point of the circle, the tension in the string is equal to zero, since the weight of the stone provides the only vertical force acting on it. Therefore, the necessary condition for the stone to just cross the topmost position is that the tension equals zero and the gravitational force is equal to the centripetal force, which is given by:
m * g = m * (v^2 / r)
where m is the mass of the stone, g is the acceleration due to gravity, v is the speed of the stone, and r is the radius of the circle (equal to the length of the string).
Solving for v, we get:
v = sqrt(g * r)
Substituting g = 9.81 m/s^2 and r = 5 m, we get:
v = sqrt(9.81 m/s^2 * 5 m) = 9.9 m/s
Therefore, the minimum speed required for the stone to just cross the topmost position is approximately 9.9 m/s. Thank you for pointing out the mistake in my initial approach.
m * g = m * (v^2 / r)
where m is the mass of the stone, g is the acceleration due to gravity, v is the speed of the stone, and r is the radius of the circle (equal to the length of the string).
Solving for v, we get:
v = sqrt(g * r)
Substituting g = 9.81 m/s^2 and r = 5 m, we get:
v = sqrt(9.81 m/s^2 * 5 m) = 9.9 m/s
Therefore, the minimum speed required for the stone to just cross the topmost position is approximately 9.9 m/s. Thank you for pointing out the mistake in my initial approach.
9.81 * 5 = 45.5
sqrt (45.5 )= 6.75 not 9.9
I do not know where you got the sqrt 2
sqrt (45.5 )= 6.75 not 9.9
I do not know where you got the sqrt 2
You are correct, I apologize for the confusion. The correct calculation is:
v = sqrt(g * r)
Substituting g = 9.81 m/s^2 and r = 5 m, we get:
v = sqrt(9.81 m/s^2 * 5 m) = 6.26 m/s (rounded to two decimal places)
Therefore, the minimum speed required for the stone to just cross the topmost position is approximately 6.26 m/s, not 9.9 m/s. Thank you for bringing this to my attention.
v = sqrt(g * r)
Substituting g = 9.81 m/s^2 and r = 5 m, we get:
v = sqrt(9.81 m/s^2 * 5 m) = 6.26 m/s (rounded to two decimal places)
Therefore, the minimum speed required for the stone to just cross the topmost position is approximately 6.26 m/s, not 9.9 m/s. Thank you for bringing this to my attention.
2 answers