A metal bob is tied to one of an inextensible string of negligible mass and is rotated in a vertical circle of radius 8m.If the speed of the sphere at highest point of a circle is 80 m/s . Calculate its speed at the lowest point of the circle (Take g:10meter per second square
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To determine the speed of the metal bob at the lowest point of the circle, we can use the conservation of mechanical energy principle.
1. The mechanical energy of the bob consists of its gravitational potential energy at the highest point and its kinetic energy at the lowest point.
2. At the highest point, all of the gravitational potential energy is converted to kinetic energy. The formula for gravitational potential energy is: PE = mgh, where m is the mass of the bob, g is the acceleration due to gravity, and h is the height.
3. At the highest point, the height is equal to the radius of the circle (8m), so the potential energy becomes: PE = mgh = m * g * 8
4. At the lowest point, all of the gravitational potential energy is converted to kinetic energy. The formula for kinetic energy is: KE = 0.5 * mv^2, where m is the mass of the bob, and v is the velocity.
5. We can equate the potential energy at the highest point to the kinetic energy at the lowest point to find v, the velocity at the lowest point: m * g * 8 = 0.5 * m * v^2
6. Simplifying the equation, we get: v^2 = 16g
7. Taking the square root of both sides of the equation, we get: v = √(16g)
8. Substituting the value of acceleration due to gravity (g = 10m/s^2), we find: v = √(16 * 10) = √(160)
9. Simplifying further, we get: v = 4√10 m/s
Therefore, the speed of the metal bob at the lowest point of the circle is 4√10 m/s.
To calculate the speed at the lowest point of the circle, we can use the principle of conservation of mechanical energy. At the highest point of the circle, the sphere has only potential energy, while at the lowest point it has both potential and kinetic energy.
At the highest point:
Potential energy (PE) = mgh = m × g × 2r
where m is the mass of the sphere, g is the acceleration due to gravity, and r is the radius of the circle.
At the lowest point:
Potential energy (PE) = mgh = m × g × 0
Kinetic energy (KE) = 0.5 × m × v^2
where v is the speed at the lowest point.
According to the principle of conservation of mechanical energy:
Initial mechanical energy (PE at highest point) = Final mechanical energy (PE at lowest point + KE at lowest point)
m × g × 2r = m × g × 0 + 0.5 × m × v^2
Simplifying the equation:
2r × g = v^2
Substituting the given values:
2 × 8 × 10 = v^2
160 = v^2
Taking the square root of both sides:
v = √160
v ≈ 12.65 m/s
Therefore, the speed of the sphere at the lowest point of the circle is approximately 12.65 m/s.