Question

A 0.50-kg ball that is tied to the end of a 1.5-m light cord is revolved in a horizontal plane, with the cord making a 30-degree angle with the vertical. If the cord can withstand a maximum tension of 9.8 N, what is the highest speed at which the ball can move?

Answer 1

To find the highest speed at which the ball can move, we can use the concept of centripetal force and the maximum tension the cord can withstand. By setting the tension equal to the maximum value, we can solve for the highest speed. The highest speed is approximately 19.6 m/s.

Step-by-step explanation:

To find the highest speed at which the ball can move, we need to consider the maximum tension the cord can withstand. In this case, the maximum tension is given as 9.8 N. We can use the concept of centripetal force to calculate the highest speed:

The centripetal force is provided by the tension in the cord, given by T = mat2, where m = mass of the ball and t = tangential speed of the ball. At the highest speed, the tension is at its maximum value:

Tmax = matmax2

Substituting the known values, m = 0.50 kg and Tmax = 9.8 N:

9.8 N = (0.50 kg)(tmax2)

Solving for tmax, we find:

tmax = √(9.8 N / 0.50 kg) = √19.6 m/s

Therefore, the highest speed at which the ball can move is approximately 19.6 m/s.

Answer 2

The highest speed at which the ball can move is approximately 7.06 m/s.

Step-by-step explanation:

To find the highest speed at which the ball can move, we need to determine the maximum tension the cord can withstand. The maximum tension occurs when the cord makes a 30-degree angle with the vertical, which is the given angle in the problem. We can calculate the maximum tension using the formula: Maximum tension = weight of the ball × (1 / sin(angle)). Substituting the values, we get: Maximum tension = 0.50 kg × (9.8 m/s²) × (1 / sin(30°)). This gives us a maximum tension of 56.3 N.

The highest speed at which the ball can move is when the tension in the cord equals the centripetal force required to keep the ball moving in a circle. The centripetal force is given by the formula: Centripetal force = (mass of the ball × velocity²) / radius. Since the radius is not given, we can assume it is the length of the light cord, which is 1.5 m. Substituting the values, we can rearrange the formula to solve for velocity: velocity = √((centripetal force × radius) / mass of the ball). Plugging in the values, we get: velocity = √((56.3 N × 1.5 m) / 0.50 kg). The highest speed at which the ball can move is approximately 7.06 m/s.