A 3.02 kg mass attached to a light string
rotates on a horizontal, frictionless table. The
radius of the circle is 1.2 m, and the string can
support a mass of 33.8 kg before breaking.
The acceleration of gravity is 9.8 m/s
2
.
What maximum speed can the mass have
before the string breaks?
Answer in units of m/s
Breaking strength
T=mg=33.8kg*9.8 N
mass = 3.02 kg
radius = 1.2 m
velocity = v
Tension on string
S=mv²/r=3.02*v²/1.2
At breaking point, S=T
3.02*v²/1.2 = 33.8*9.8
Solve for v. I get 11.5 m/s
Why did the string go to the therapist? Because it was feeling a little stretched out!
To find the maximum speed, we can use the concept of centripetal force. The centripetal force is equal to the tension force in the string. If the tension force exceeds the maximum tension the string can handle, it will break.
The centripetal force can be calculated using the equation: fc = m * ac, where m is the mass of the object and ac is the centripetal acceleration.
The centripetal acceleration can be calculated using the equation: ac = (v^2) / r, where v is the velocity and r is the radius.
Since we want to find the maximum velocity before the string breaks, we need to find the maximum tension force. Given that the string can support a mass of 33.8 kg before breaking, the maximum tension force can be found using the formula: T_max = m_max * g, where m_max is the maximum mass the string can support and g is the acceleration due to gravity.
Now, let's put it all together:
T_max = 33.8 kg * 9.8 m/s^2 = 331.24 N
We can equate this maximum tension force with the centripetal force:
m * ac = T_max
Substituting the values we know:
3.02 kg * (v^2)/1.2 m = 331.24 N
Simplifying:
(v^2)/1.2 = 109.6026 m^2/s^2
v^2 = 131.5231 m^2/s^2
Taking the square root of both sides:
v = 11.47 m/s
So, the maximum speed the mass can have before the string breaks is approximately 11.47 m/s. Keep an eye on that speedometer!
To find the maximum speed that the mass can have before the string breaks, we can use the concept of centripetal force.
The centripetal force required to keep the mass moving in a circle is provided by the tension in the string. At the maximum speed before the string breaks, the tension in the string will be equal to the breaking strength of the string.
The breaking strength of the string can be determined by the maximum mass it can support, which is given as 33.8 kg.
Let's denote the maximum speed of the mass as "v". At this speed, the tension in the string will be equal to the weight of the maximum mass that the string can support.
The weight of the maximum supported mass is calculated as follows:
Weight = mass * acceleration due to gravity
Weight = 33.8 kg * 9.8 m/s^2
Now, the tension in the string is also the centripetal force required to keep the mass moving in a circle. The centripetal force can be calculated using the following formula:
Centripetal force = mass * (velocity)^2 / radius
Setting these two forces equal to each other, we have:
Tension = Centripetal force
33.8 kg * 9.8 m/s^2 = 3.02 kg * v^2 / 1.2 m
Solving for v^2, we have:
v^2 = (33.8 kg * 9.8 m/s^2 * 1.2 m) / 3.02 kg
Now, we can find v by taking the square root of both sides:
v = sqrt((33.8 kg * 9.8 m/s^2 * 1.2 m) / 3.02 kg)
Calculating this expression, we get:
v = sqrt(418.32 m^2/s^2) = 20.46 m/s
Therefore, the maximum speed that the mass can have before the string breaks is approximately 20.46 m/s.
To find the maximum speed that the mass can have before the string breaks, we need to consider the tension in the string at this maximum speed.
Since the system is rotating in a circle, there must be a centripetal force acting towards the center of the circle. In this case, the tension in the string provides the centripetal force.
The centripetal force can be calculated using the formula:
F_centripetal = m * a_centripetal
where F_centripetal is the centripetal force, m is the mass, and a_centripetal is the centripetal acceleration.
The centripetal acceleration can be calculated using the formula:
a_centripetal = v^2 / r
where v is the velocity and r is the radius of the circle.
Now, we can equate the centripetal force to the tension in the string:
F_tension = F_centripetal
where F_tension is the tension in the string.
To find the maximum speed before the string breaks, we need to find the tension in the string when it is at its maximum limit. We are given that the string can support a mass of 33.8 kg before breaking. Therefore, the maximum tension in the string is equal to the weight of this mass:
F_tension(max) = m(max) * g
where m(max) is the maximum mass the string can support and g is the acceleration due to gravity.
Now we can set up the equations:
F_tension = m * a_centripetal
m(max) * g = m * a_centripetal
Substituting the values given in the question:
33.8 kg * 9.8 m/s^2 = 3.02 kg * (v^2 / r)
Simplifying the equation:
v^2 = (33.8 kg * 9.8 m/s^2 * r) / 3.02 kg
v^2 = 1096.06 m^2/s^2 * r
Taking the square root of both sides:
v = √(1096.06 m^2/s^2 * r)
v = √(1096.06 m^2/s^2 * 1.2 m)
v = √(1315.272 m^2/s^2)
v ≈ 36.24 m/s
Therefore, the maximum speed the mass can have before the string breaks is approximately 36.24 m/s.