a body is moving in a vertical circle such that the velocities of body at different points are critical the ratio of velocities of body at angular displacements 60° 120° from lowest point is √2:1
How the hell its wrong!!!
It's perfectly explained!
Anyway thanks for that dude who posted the right one!!!!!
To solve this problem, let's consider the motion of a body moving in a vertical circle. The key concept to understand here is that of centripetal force, which is responsible for keeping the body in circular motion.
At the highest point of the circle, the body is momentarily at rest, and its velocity is zero. As the body moves downwards, the velocity increases due to the gravitational force acting on it. At some point, when the body reaches the bottom of the circle, its velocity is at its maximum value.
Now, let's focus on the specific angular displacements mentioned (60° and 120° from the lowest point). We need to find the ratio of the velocities at these two points.
To approach this problem, we can use the conservation of energy principle. At any point in the circular path, the sum of kinetic energy (KE) and potential energy (PE) remains constant. In terms of velocities, we can express this as:
KE + PE = constant
Since the body is moving in a vertical circle, we can express the potential energy in terms of height (h) relative to the lowest point. At the lowest point, the height is zero, so the potential energy is also zero. Thus, the expression becomes:
KE + 0 = constant
KE = constant
Now, from the given ratio (√2:1), we can write:
velocity at 60° = √2 x velocity at 120°
Let's assign a variable for the velocity at the lowest point, such as v_0.
Using this, we can write the following equations:
KE at 60° = (1/2)mv^2 = (1/2)mv_0^2
KE at 120° = (1/2)mv^2 = (1/2)m(√2v_0)^2 = (1/2)m2v_0^2 = mv_0^2
Since the kinetic energy at the highest and lowest points should be the same (due to conservation of energy), we can equate the two equations:
(1/2)mv_0^2 = mv_0^2
Canceling the common terms, we get:
1/2 = 1
This equation cannot be true, which indicates that there might be an error in the given problem statement.
Please check the problem again and provide any additional information if available.
a body is moving in a vertical circle such that the velocities of body at different points are critical the ratio of velocities of body at angular displacements 60° 120° from lowest point is
And =√2:1
height h at angle T = R - R cos T
height h at 60 deg = R/2
height h at 120 deg = 3 R/2
kinetic energy
(1/2)mv^2 + m g h = constant
at 60 deg
(1/2)v1^2 + g(R/2)=(1/2)v2^2 + g(3R/2)
(v1^2 -v2^2) = 2 g R
if it just makes it to the top then
(1/2) m v1^2 = m g * 2R
v1^2 = 4 g R
(4 g R - v2^2 ) = 2 g R
v2^2 = 2 g R
v1^2/v2^2 = 4 g R / 2 g R
V1/V2 = sqrt 2 / 1