a .45kg ball, attached to the end of a horizontal cord, is rotated in a circle of radius 1.3m on a frictionless horizontal surface. if the cord will break when the tension in it exceeds 75 N, what is the maximum speed the ball can have?

The answer is actually 14.72 m/s :)

(the square root of 1.3*75/.45 = 14.72)

75=mv^2/r

solve for v.

Well, isn't this a swingin' question! So, we have a ball on a cord, swirling around in a circle. The tension in the cord can't exceed 75 N or else it'll snap like a rubber band on a catapult. Now, to find the maximum speed, we need to look at the forces at play.

When an object moves in a circle, it experiences a centripetal force pulling it towards the center. In this case, the tension in the cord provides that force. So we can use the equation: Tension = (mass x velocity^2) / radius.

To find the maximum speed, we need to make the tension equal to 75 N. Rearranging the equation, we get: velocity^2 = (tension x radius) / mass.

Plugging in the values, we have: velocity^2 = (75 N x 1.3 m) / 0.45 kg.

Now, let's crunch some numbers... well, my "number cruncher" seems to be on a lunch break. But fear not! I have my handy-dandy calculator. After a bit of calculating, I find that the maximum speed the ball can have is approximately 11.67 m/s.

So, make sure that ball doesn't get too fast, or it might take the cord for a spin it can't handle!

To find the maximum speed of the ball, we need to determine the maximum tension in the cord when the ball is at its maximum speed.

We can start by using the centripetal force equation:

Fc = m * v^2 / r

Where:
Fc is the centripetal force,
m is the mass of the ball (0.45 kg),
v is the velocity of the ball,
r is the radius of the circular path (1.3 m).

Since the tension in the cord provides the centripetal force, we can set the tension (T) equal to Fc:

T = Fc

Therefore:

T = m * v^2 / r

To find the maximum tension, we set T equal to the breaking point (75 N):

75 N = (0.45 kg) * v^2 / (1.3 m)

Now, we can solve for the maximum velocity (v):

75 N * 1.3 m = 0.45 kg * v^2
97.5 N * m = 0.45 kg * v^2
v^2 = (97.5 N * m) / (0.45 kg)
v^2 ≈ 216.67 N * m / kg

Taking the square root of both sides:

v ≈ √(216.67 N * m / kg)
v ≈ 14.72 m/s

Therefore, the maximum speed the ball can have is approximately 14.72 m/s.

To find the maximum speed of the ball, we need to determine the maximum tension in the cord when the ball is rotating at that speed. Let's go step by step:

1. The tension in the cord is equal to the centripetal force required to keep the ball moving in a circle. This force can be calculated using the formula:

Tension = mass × centripetal acceleration

2. The centripetal acceleration can be calculated using the formula:

Centripetal acceleration = (velocity squared) / radius

3. Therefore, we can rewrite the formula for the tension as:

Tension = mass × (velocity squared) / radius

4. We know the mass of the ball is 0.45 kg, and the radius of the circle is 1.3 m.

5. The question states that the tension in the cord can't exceed 75 N. We can set up the inequality:

Tension ≤ 75 N

6. Substituting the values we know into the equation from step 3, we get:

75 N ≥ (0.45 kg) × (velocity squared) / (1.3 m)

7. Rearranging the equation, we can solve for the maximum velocity:

(0.45 kg) × (velocity squared) / (1.3 m) ≤ 75 N

8. Multiplying both sides of the inequality by (1.3 m), we get:

(0.45 kg) × (velocity squared) ≤ 75 N × (1.3 m)

9. Dividing both sides of the equation by (0.45 kg), we can solve for the velocity squared:

velocity squared ≤ (75 N × 1.3 m) / (0.45 kg)

10. Taking the square root of both sides of the equation gives us the maximum velocity:

velocity ≤ √((75 N × 1.3 m) / (0.45 kg))

11. Calculating the expression on the right-hand side will give us the final answer, which is the maximum speed the ball can have.

Note: Make sure to use a calculator or a software tool to perform the calculations accurately.