A 3.21 kg mass attached to a light string
rotates on a horizontal, frictionless table. The
radius of the circle is 0.855 m, and the string
can support a mass of 15.3 kg before breaking.
What maximum speed can the mass have
before the string breaks?
Answer in units of m/s
m₁=3.21 kg
m₂=15.3 kg
m₂g=T
m₁v²/R=T
m₁v²/R= m₂g
v=sqrt{ m₂gR/ m₁}=
=sqrt{15.3•9.8•0.855/3.21} = 1.8 m/s
6.32 m/s
To find the maximum speed at which the mass can rotate before the string breaks, we need to consider the tension in the string.
The tension in the string is the centripetal force required to keep the mass in circular motion.
We can calculate the tension using the centripetal force formula:
Tension = (Mass × Velocity²) / Radius
Given:
Mass (m) = 3.21 kg
Radius (r) = 0.855 m
Let's assume the maximum speed before the string breaks is V_max.
Tension (T) = [(3.21 kg) × (V_max²)] / 0.855 m
Now, we need to find the maximum speed at which the tension in the string reaches its maximum limit of 15.3 kg.
T = 15.3 kg
[(3.21 kg) × (V_max²)] / 0.855 m = 15.3 kg
Solve for V_max:
[(3.21 kg) × (V_max²)] = (15.3 kg) × (0.855 m)
V_max² = [(15.3 kg) × (0.855 m)] / (3.21 kg)
V_max² = 4.08525 m²/s²
V_max = √(4.08525 m²/s²)
V_max ≈ 2.0217 m/s
Therefore, the maximum speed the mass can have before the string breaks is approximately 2.0217 m/s.
To find the maximum speed at which the mass can rotate before the string breaks, we need to consider the tension in the string at that speed.
The tension in the string provides the centripetal force required to keep the mass moving in a circle. At maximum speed, the centripetal force is at its highest and equal to the maximum tension the string can withstand before breaking.
The centripetal force can be calculated using the following equation:
Fc = m * v^2 / r
Where Fc is the centripetal force, m is the mass, v is the velocity, and r is the radius.
Assuming the maximum tension the string can support is equal to the weight of the attached mass, we can set up an equation:
m_max * g = m * v^2 / r
Where m_max is the maximum mass the string can support before breaking, g is the acceleration due to gravity (approximately 9.8 m/s^2), m is the mass of the rotating object, v is the maximum velocity, and r is the radius.
Rearranging the equation to solve for v:
v^2 = r * g * m_max / m
Substituting the known values:
v^2 = 0.855 m * 9.8 m/s^2 * 15.3 kg / 3.21 kg
v^2 = 40.95 m^2/s^2
Finally, taking the square root of both sides to solve for v:
v = sqrt(40.95) m/s
v ≈ 6.4 m/s
Therefore, the maximum speed the mass can have before the string breaks is approximately 6.4 m/s.