A spring has a force constant of 478 N/m and an unstretched length of 9 cm. one end is attached to a post that is free to rotate in the center of a smooth table. the other end is attached to a 4kg disc moving in uniform circular motion on the table, which stretches the spring by 3 cm. Note: Friction is negligible.
1.What is the centripetal force Fc on the disk? Answer in units of N.
2.What is the work done on the disk by the spring during one full circle?
Is the 4kg disk sliding or rolling around the table?
If sliding,
cenripetal force=spring force= Kx=478N*.03
b) work is zero. Force is not in the direction of motion.
It's sliding.... Thank You very much!!!
To find the answers to the given questions, we'll break them down step-by-step.
1. The centripetal force, Fc, is the force that keeps an object moving in a circular path. It is given by the equation:
Fc = m * v^2 / r
where m is the mass of the object, v is the velocity, and r is the radius.
In this case, the mass of the object is 4 kg, and the radius is the stretched length of the spring, which is the sum of the unstretched length (9 cm) and the amount the spring is stretched (3 cm). So the radius is 9 cm + 3 cm = 12 cm = 0.12 m.
Since friction is negligible, the force of the spring, Fspring, must provide the centripetal force. Thus, Fc = Fspring.
Now, to find the force of the spring, we can use Hooke's Law, which relates the force exerted by a spring to its displacement:
Fspring = k * x
where k is the force constant of the spring and x is the displacement.
In this case, the force constant of the spring is given as 478 N/m, and the displacement is 3 cm = 0.03 m.
Substituting these values into the equation, we can calculate the centripetal force:
Fc = Fspring = k * x = 478 N/m * 0.03 m = 14.34 N
Thus, the centripetal force on the disk is 14.34 N.
2. The work done on the disk by the spring during one full circle can be calculated by integrating the force exerted by the spring over the displacement of the disk.
The work done, W, is given by the equation:
W = ∫ Fspring * dx
where Fspring is the force exerted by the spring and dx is the differential displacement.
The force exerted by the spring is constant throughout the displacement of the disk, so the equation simplifies to:
W = Fspring * ∆x
where ∆x is the total displacement of the disk in one full circle.
Since the disk is moving in uniform circular motion, its total displacement in one full circle is equal to the circumference of the circle, 2πr.
The radius of the circle, as previously calculated, is 0.12 m.
Therefore, the total displacement of the disk is 2π * 0.12 m = 0.24π m.
Substituting the values into the equation, we can calculate the work done:
W = Fspring * ∆x = 478 N/m * 0.24π m ≈ 361.81 N.m (rounded to two decimal places)
Thus, the work done on the disk by the spring during one full circle is approximately 361.81 N.m.
To find the centripetal force on the disk, we can use the equation:
Fc = m * ω² * r
where Fc is the centripetal force, m is the mass of the disk, ω (omega) is the angular velocity, and r is the radius of the circular motion.
1. First, let's find the angular velocity (ω):
The angular velocity (ω) can be calculated using the equation:
ω = v / r
where v is the linear velocity of the disk. Since the motion is uniform circular motion, the linear velocity is given by:
v = 2πr / T
where T is the period of one full circle.
2. To find the period (T), we can use the formula:
T = 2π / f
where f is the frequency of the circular motion. We can find the frequency using the fact that the spring is stretched by 3 cm. Since the unstretched length of the spring is 9 cm, the elongation of the spring is 3 cm. This means that the frequency (f) is given by:
f = k / (2πm)
where k is the force constant of the spring and m is the mass of the disk.
Once we have the value of f, we can substitute it in the equation for T to find the value of T.
3. Now that we know the angular velocity (ω), we can find the centripetal force (Fc):
Fc = m * ω² * r
where m is the mass of the disk and r is the radius of the circular motion.
To find the work done by the spring during one full circle, we can use the equation:
W = ½ * k * x²
where W is the work done by the spring, k is the force constant of the spring, and x is the elongation or compression of the spring.
Let's calculate the answers to the questions using the given values:
Given:
Force constant of the spring (k) = 478 N/m
Unstretched length of the spring (x_0) = 9 cm
Stretched length of the spring (x) = 3 cm
Mass of the disk (m) = 4 kg
1. Calculating the centripetal force (Fc):
First, calculate the frequency (f):
f = k / (2πm)
f = 478 / (2π * 4)
Now, calculate the period (T):
T = 2π / f
Next, calculate the angular velocity (ω):
ω = 2π / T
Finally, calculate the centripetal force (Fc):
Fc = m * ω² * r
2. Calculating the work done by the spring (W):
W = ½ * k * x²
Substituting the given values for k and x:
W = 0.5 * 478 * (3/100)²
Now, calculate the values of Fc and W using the steps outlined above to find the final answers.