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A particle describes a horizontal circle in a conical funnel whose inner surface is smooth with speed of 0.5 m/s. What is the height of the plane of the circle from the vertex of the funnel?
- a)0.25 cm
- b)2 cm
- c)4 cm
- d)2.5 cm
Correct answer is option 'D'. Can you explain this answer?
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A particle describes a horizontal circle in a conical funnel whose in...
To solve this problem, we can use the concept of centripetal acceleration. The particle moving in a horizontal circle experiences a centripetal acceleration towards the center of the circle. This centripetal acceleration is provided by the gravitational force acting on the particle.
Let's break down the problem step by step:
1. Identify the forces acting on the particle:
- Gravitational force (mg) acting vertically downwards
- Normal force (N) acting perpendicular to the surface of the funnel
- Centripetal force (F_c) acting towards the center of the circle
2. Determine the components of the gravitational force:
- The gravitational force can be resolved into two components: one parallel to the surface of the funnel (mg*sinθ) and one perpendicular to the surface of the funnel (mg*cosθ), where θ is the angle between the surface of the funnel and the vertical.
3. Determine the components of the normal force:
- The normal force can also be resolved into two components: one parallel to the surface of the funnel (N*sinθ) and one perpendicular to the surface of the funnel (N*cosθ).
4. Equate the centripetal force with the horizontal component of the gravitational force:
- F_c = mg*sinθ
- mv^2/r = mg*sinθ
- v^2/r = g*sinθ
5. Calculate the height of the plane of the circle from the vertex of the funnel:
- The height (h) can be determined using the right triangle formed by the height, the radius of the circle (r), and the slant height of the funnel (l).
- sinθ = h/l
- h = l*sinθ
6. Substitute the value of sinθ from the equation in step 4:
- h = l*(v^2)/(g*r)
7. Calculate the height:
- Substituting the given values:
- v = 0.5 m/s
- r = l (as the circle is on the surface of the funnel)
- g = 9.8 m/s^2
- h = l*(0.5^2)/(9.8*l)
- h = 0.5/9.8
- h ≈ 0.051 m or 5.1 cm
Therefore, the height of the plane of the circle from the vertex of the funnel is approximately 5.1 cm. None of the given options match this value, so it seems there may be an error in the options provided.
Let's break down the problem step by step:
1. Identify the forces acting on the particle:
- Gravitational force (mg) acting vertically downwards
- Normal force (N) acting perpendicular to the surface of the funnel
- Centripetal force (F_c) acting towards the center of the circle
2. Determine the components of the gravitational force:
- The gravitational force can be resolved into two components: one parallel to the surface of the funnel (mg*sinθ) and one perpendicular to the surface of the funnel (mg*cosθ), where θ is the angle between the surface of the funnel and the vertical.
3. Determine the components of the normal force:
- The normal force can also be resolved into two components: one parallel to the surface of the funnel (N*sinθ) and one perpendicular to the surface of the funnel (N*cosθ).
4. Equate the centripetal force with the horizontal component of the gravitational force:
- F_c = mg*sinθ
- mv^2/r = mg*sinθ
- v^2/r = g*sinθ
5. Calculate the height of the plane of the circle from the vertex of the funnel:
- The height (h) can be determined using the right triangle formed by the height, the radius of the circle (r), and the slant height of the funnel (l).
- sinθ = h/l
- h = l*sinθ
6. Substitute the value of sinθ from the equation in step 4:
- h = l*(v^2)/(g*r)
7. Calculate the height:
- Substituting the given values:
- v = 0.5 m/s
- r = l (as the circle is on the surface of the funnel)
- g = 9.8 m/s^2
- h = l*(0.5^2)/(9.8*l)
- h = 0.5/9.8
- h ≈ 0.051 m or 5.1 cm
Therefore, the height of the plane of the circle from the vertex of the funnel is approximately 5.1 cm. None of the given options match this value, so it seems there may be an error in the options provided.
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Community Answer
A particle describes a horizontal circle in a conical funnel whose in...
The particle is moving in circular path
. . . (ii)
From the figure, mg = R sin θ . . . (i)
. . . (ii)From equations (i) and (ii) we get
∴ 
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