A block of mass M=2 kg is swung on a rope in a veritcal circle(the direction of motion is vertical to the ground) of radius r. When the block is at the top of the circle, the tension in the rope is measured to be 10N. What is the tension in the rope when the block is at the bottom of the circle?
If the speed at the top is v1, then the tension there is:
T1 = m [v1^2/r - g]
It is given that T1 = 10 N
At the bottom the tension will be:
T2 = m [v2^2/r + g]
where v2 is the speed at the bottom.
Using conservation of energy, you find:
1/2 v2^2 - 1/2 v1^2 = 2 r g ---->
v2^2/r - v1^2/r = 4 g
If you subtract the equation for T2 and T1 and insert the above relation between v2 and v1, you get:
T2 - T1 = 6 m g
127.6
To find the tension in the rope when the block is at the bottom of the circle, we can use the concept of centripetal force.
When the block is at the top of the circle, the tension in the rope provides the centripetal force required to keep the block moving in a circular path. At the top of the circle, the tension force and the gravitational force are acting in the same direction, so we can write the equation:
Tension + Weight = Centripetal Force
Where:
Tension is the tension in the rope (unknown).
Weight is the weight of the block, which is equal to M * g, where M is the mass of the block (2 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Centripetal Force is equal to M * v^2 / r, where M is the mass of the block, v is the velocity of the block, and r is the radius of the circle.
Since the block is at the top of the circle, its velocity is zero because it momentarily comes to a stop at the highest point of its motion. Therefore, the centripetal force is also zero at this point.
So, we can rewrite the equation as:
Tension + M * g = 0
Solving for Tension:
Tension = - M * g
Now, when the block is at the bottom of the circle, the tension in the rope provides the centripetal force required to keep the block moving in a circular path. At the bottom of the circle, the tension force and the gravitational force are acting in opposite directions, so we can write the equation:
Tension - Weight = Centripetal Force
Using the same values for Weight and Centripetal Force as before, we can solve for Tension:
Tension - M * g = M * v^2 / r
Since the velocity at the bottom of the circle can be determined from the conservation of energy, which states that the sum of potential and kinetic energy remains constant, we can write:
Potential Energy at the top = Kinetic Energy at the bottom
M * g * h = 1/2 * M * v^2
Given the height (h) of the circle is equal to 2 * r (since the radius is r), we can substitute this value into the equation:
M * g * (2 * r) = 1/2 * M * v^2
Simplifying the equation:
2 * M * g * r = 1/2 * M * v^2
Solving for v:
v^2 = 4 * g * r
v = sqrt(4 * g * r)
Now, substitute this value of v back into the equation for tension at the bottom:
Tension - M * g = M * (4 * g * r) / r
Tension - M * g = 4 * M * g
Tension = 5 * M * g
Substituting the values of M (2 kg) and g (9.8 m/s^2), we can finally calculate the tension at the bottom of the circle:
Tension = 5 * 2 kg * 9.8 m/s^2
Tension = 98 N
Therefore, the tension in the rope when the block is at the bottom of the circle is 98 N.