a .45 kg ball attached to the end of a horizontal cord is rotated in a circle of radius 1.3 m on a frictionless horizontal surface. if the cord will break when the tension in it exceeds 75N what is the maximum speed the ball can have?
To find the maximum speed the ball can have before the cord breaks, we need to consider the tension in the cord. The tension in the cord is equal to the centripetal force required to keep the ball moving in a circle.
The centripetal force is given by the formula:
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)
Given that the tension in the cord should not exceed 75 N, we can set Fc equal to 75 N and solve for v:
75 = (0.45 * v^2) / 1.3
To solve for v, we can rearrange the equation:
v^2 = (75 * 1.3) / 0.45
v^2 = 216.67
Taking the square root of both sides:
v = √(216.67)
v ≈ 14.72 m/s
Therefore, the maximum speed the ball can have before the cord breaks is approximately 14.72 m/s.
To find the maximum speed the ball can have without breaking the cord, we need to use the concept of centripetal force.
Here's how you can calculate the maximum speed:
1. First, let's find the maximum tension in the cord when the ball is rotating at its maximum speed without breaking the cord. We know the maximum tension is 75 N.
2. The tension in the cord acts as the centripetal force, which keeps the ball in circular motion. Therefore, we can equate the maximum tension to the centripetal force:
Tension (F) = Centripetal Force (Fc)
T = m * a
where:
T = tension in the cord
m = mass of the ball
a = centripetal acceleration
3. The centripetal acceleration (a) can be calculated using the formula:
a = v^2 / r
where:
a = centripetal acceleration
v = velocity or speed of the ball
r = radius of the circle
4. Now, substitute the value of the centripetal acceleration into the tension equation:
T = m * (v^2 / r)
5. Rearrange the equation to solve for v:
v = sqrt((T * r) / m)
6. Plug in the given values into the equation to calculate the maximum speed:
v = sqrt((75 N * 1.3 m) / 0.45 kg)
v ≈ sqrt(243 / 0.45)
v ≈ sqrt(540)
v ≈ 23.24 m/s
Therefore, the maximum speed the ball can have without breaking the cord is approximately 23.24 m/s.