A ball of mass 0.750 kg is attached to the end of a 1.20m long cord that can withstand a maximum tension of 50.0 N. If swung in a vertical circle, what is the maximum velocity the ball can achieve before the string breaks?
Since the circle is vertical, maximum tension is experienced when the ball is at the bottom of the loop. Tension must support both weight and centripetal force.
Require that
M*V^2/R + M*g = (maximum tension) Tmax
= 50.0 N
Then solve for V.
M*V^2/R = 50 - 7.35 = 42.65 N
V = sqrt(42.65*1.2/0.75)= 8.26 m/s
To determine the maximum velocity the ball can achieve before the string breaks, we need to consider the forces acting on the ball at the maximum tension.
In this scenario, the ball is being swung in a vertical circle, so at the bottom of the circle, the tension in the string will be an additional force acting downward along with the weight of the ball. At the top of the circle, the tension will be acting upward, opposing the weight of the ball.
At the bottom of the circle:
The net force is the difference between the tension and the weight of the ball. We can calculate the tension at the bottom using the maximum tension the cord can withstand:
Tension at bottom = Maximum tension - Weight of the ball
The weight of the ball can be calculated using the formula:
Weight = mass * acceleration due to gravity
At the top of the circle:
The net force is the sum of the tension and the weight of the ball. We can calculate the tension at the top using the formula:
Tension at top = Weight of the ball + Maximum tension
Now, to find the maximum velocity, we can equate the centripetal force at the maximum tension to the net force at the bottom of the circle.
Centripetal force = net force at the bottom
The centripetal force in this case is given by:
Centripetal force = mass of the ball * (velocity)^2 / radius
Rearranging the equations, we have:
Velocity^2 = (Centripetal force * radius) / mass of the ball
Substituting the expressions for the centripetal force and the tension at the bottom, we get:
Velocity^2 = ((Maximum tension - Weight of the ball) * radius) / mass of the ball
Finally, taking the square root of both sides of the equation gives us:
Velocity = √(((Maximum tension - Weight of the ball) * radius) / mass of the ball)
Now we can calculate the maximum velocity.