A 0.75-kg ball is attached to a 1.0-m rope and whirled in a vertical circle. The rope will break when the tension exceeds 450 N. What is the maximum speed the ball can have at the bottom of the circle without breaking the rope?

max tension is at the bottom

tensionmax=mg+mv^2/r solve for v

To find the maximum speed the ball can have at the bottom of the circle without breaking the rope, we need to consider the forces acting on the ball at that point.

First, let's consider the forces acting on the ball when it is at the bottom of the circle:

1. Weight (mg): The weight of the ball acts downward and is given by the equation:
Weight = mass * acceleration due to gravity = (0.75 kg) * (9.8 m/s^2) = 7.35 N

2. Tension (T): The tension in the rope acts upward. At the bottom of the circle, the tension is at its maximum value, which is 450 N.

At the bottom of the circle, the net force acting on the ball is the difference between the tension and the weight:

Net Force = T - Weight

Since the ball is moving in a circular path, the net force is related to the centripetal force:

Net Force = mb * (v^2 / r),

where mb is the mass of the ball, v is the velocity of the ball, and r is the radius of the circular path.

Since the net force and the centripetal force are equal, we can set these equations equal to each other:

T - Weight = mb * (v^2 / r)

Now we can solve for the maximum velocity (v) at the bottom of the circle:

T - Weight = mb * (v^2 / r)
450 N - 7.35 N = (0.75 kg) * (v^2 / 1.0 m)

Simplifying the equation:

442.65 N = 0.75 kg * v^2

Dividing both sides by 0.75 kg:

v^2 = (442.65 N) / (0.75 kg)
v^2 = 590.2 m^2/s^2

Taking the square root of both sides:

v = sqrt(590.2 m^2/s^2)
v ≈ 24.27 m/s

Therefore, the maximum speed the ball can have at the bottom of the circle without breaking the rope is approximately 24.27 m/s.

To find the maximum speed the ball can have without breaking the rope, we can set up an equation using the forces acting on the ball at the bottom of the circle.

At the bottom of the circle, the tension in the rope must provide both the centripetal force required to keep the ball in circular motion and counteract the force of gravity.

First, let's calculate the gravitational force acting on the ball:

Force_gravity = mass * acceleration_due_to_gravity

where:
mass = 0.75 kg (given)
acceleration_due_to_gravity = 9.8 m/s^2

Force_gravity = 0.75 kg * 9.8 m/s^2
Force_gravity = 7.35 N

Next, let's calculate the centripetal force required to keep the ball in circular motion:

Centripetal_force = mass * (velocity^2) / radius

where:
mass = 0.75 kg (given)
velocity = maximum speed of the ball at the bottom of the circle (to be determined)
radius = length of the rope = 1.0 m (given)

Setting the tension in the rope equal to the sum of the gravitational force and the centripetal force, we have:

Tension = Force_gravity + Centripetal_force

Tension = 450 N (given)

Substituting the values and rearranging the equation to solve for the velocity:

450 N = 7.35 N + (0.75 kg * velocity^2) / 1.0 m

450 N - 7.35 N = 0.75 kg * velocity^2

442.65 N = 0.75 kg * velocity^2

Dividing both sides by 0.75 kg:

589.2 m^2/s^2 = velocity^2

Taking the square root of both sides to solve for velocity:

velocity = √589.2 m^2/s^2
velocity ≈ 24.26 m/s

Therefore, the maximum speed the ball can have at the bottom of the circle without breaking the rope is approximately 24.26 m/s.

T=mv^2/r + mg

or T= mg + mv^2/r
T= 450N r= 1
m= 0.75
450 = (o.75)(9.8) + (0.75)(v^2)
v^2= 450/8
v= 7.5