A mass of 0.04 kg is attached to a cord passing through a small hole in a frictionless, horizontal surface. The mass is initially orbiting with speed 4.9 m/s in a circle of radius 0.4 m. The cord is then slowly pulled from below, decreasing the radius of the circle to r2 = 0.09 m. Find the tension in the cord when the point mass is at r2.
To find the tension in the cord when the point mass is at r2, we can use the principle of conservation of angular momentum.
Angular momentum (L) of an object rotating around an axis is given by the formula:
L = mvr
where:
- m is the mass of the object
- v is the velocity of the object
- r is the radius of the circle
Since no external torque is acting on the system, the angular momentum of the object remains constant. Therefore, we can write:
L1 = L2
where:
L1 is the initial angular momentum at radius r1
L2 is the final angular momentum at radius r2
We can express the angular momentum at each radius as:
L1 = m1 * v1 * r1
L2 = m2 * v2 * r2
where:
m1 and m2 are the masses at r1 and r2, respectively
v1 and v2 are the velocities at r1 and r2, respectively
r1 is the initial radius (0.4 m)
r2 is the final radius (0.09 m)
Given:
m = 0.04 kg
v1 = 4.9 m/s
r1 = 0.4 m
r2 = 0.09 m
We need to find the tension in the cord, which is the centripetal force acting on the mass at r2.
The centripetal force (Fc) acting on an object moving in a circle is given by:
Fc = m * a
where:
m is the mass of the object
a is the centripetal acceleration
The centripetal acceleration (a) can be calculated as:
a = (v2^2) / r2
where:
v2 is the velocity at r2
r2 is the radius
Given:
m = 0.04 kg
v2 = (L2 / (m2 * r2))
r2 = 0.09 m
Now, let's calculate the angular momentum at each radius:
L1 = m1 * v1 * r1
L2 = m2 * v2 * r2
Given:
m1 = m
v1 = 4.9 m/s
r1 = 0.4 m
L1 = m * v1 * r1
Next, let's calculate the velocity at r2:
v2 = (L2 / (m2 * r2))
Finally, let's calculate the centripetal force:
Fc = m * a
Now you have all the information needed to find the tension in the cord when the point mass is at r2. Use the formulas and given values to solve the problem.