Question

A student attaches a mass \[M\] to a cord of length \[L\] and spins it rapidly in a horizontal circle above her head. When the speed of the mass reaches \[v_1\], the cord breaks, and the mass flies off. The student wants to increase the speed to \[4v_1\]. What minimum cord length would allow the student to be able to increase the speed to \[4v_1\]

Answer 1

To increase the mass's speed to four times its original speed without exceeding the cord's maximum tolerable force, the length of the cord must be increased to 16 times the original length. This follows from the centripetal force being proportional to the square of the velocity and inversely proportional to the radius.

Step-by-step explanation:

The question at hand involves understanding the relationship between centripetal force, the velocity at which an object is moving in a circle, and the radius (or length of the cord) of that circle. According to the centripetal force formula F = mv^2/r, where F is centripetal force, m is mass, v is velocity, and r is radius, we can determine the necessary conditions to increase the speed of the mass without breaking the cord.

When the cord length is L and the speed is v1, the centripetal force is F1 = M * v1^2/L. To increase the speed to 4v1 while ensuring that the maximum force the cord can withstand (which is equal to F1) is not exceeded, we would increase the velocity fourfold, meaning the centripetal force would increase if the radius remained the same. To keep the centripetal force the same at a higher velocity, the radius must increase. Using the formula, we can set up the following equation: M * (4v1)^2 / new L = M * v1^2 / L. Simplifying gives 16 * M * v1^2 / new L = M * v1^2 / L, and therefore new L = 16 * L.

So, to maintain the same centripetal force while quadrupling the velocity, the length of the cord needs to be 16 times the original length. Therefore, the minimum cord length that would allow the student to safely increase the speed to 4v1 is 16L.

Answer 2

To spin a mass at a speed of 4v_1 without breaking the cord, the length of the cord must be increased to 16 times its original length, assuming a direct square relationship between required centripetal force and velocity.

Step-by-step explanation:

The question asked involves the physics concept of centripetal force, which is the requisite force to keep an object moving in a circular path. The force required to keep an object of mass M moving in a horizontal circle of radius L at a given speed v is proportional to the square of the speed and inversely proportional to the radius of the circle. Since the cord breaks at speed v_1, to increase the speed to 4v_1 without breaking the cord, we must consider the increase in centripetal force required. The centripetal force is given by the formula F_centripetal = Mv^2/L. At speed 4v_1, the force is 16 times greater because force is proportional to the square of the velocity (F_centripetal ~ v^2). Hence, to prevent the cord from breaking at this higher speed, the length of the cord must provide a lower force. This would require the new cord length L_new to be 16 times the original length L, as F_centripetal ~ 1/L. This results in a new length requirement of L_new = 16L to sustain the increased speed of 4v_1.