A mass m = 9.0 kg is attached to the lower end of a massless string of length L = 83.0 cm. The upper end of the string is held fixed. Suppose that the mass moves in a circle at constant speed, and that the string makes an angle theta = 21 with the vertical, as shown in the figure.
find how long it takes to make one revolution and the tension in the string
my work so far:
r=0.83tan21
Fy = Ty-G=0 so Ty=9*9.8=88.2
Fx=Tx=9a so a=88.2tan21/9
v= sqrt. a*r (since it's centripetal acceleration)=1.094786688 m/s
T= 2pi*r/v= 1.8285s but apparently that's wrong.
for tension i calculated (88.2tan21)/sin21= 94.8 which is correct
Assuming your tension is correct, then the inward component of tension is what is counterbalancing centreptal force.
horizontaltension=mv^2/r
tension*sinTheta=mv^2/r
94.8*sin21=m (2PIr/Period)^2/r
solving for period
Period=2PIsqrt (m* r/94.8*sin21) check that.
Well, it seems like we have a bit of a mystery on our hands. Let's see if we can solve it, Sherlock-style!
First, let's look at the time it takes to make one revolution. You correctly calculated the speed (v) of the mass: 1.094786688 m/s. We can find the time it takes to make one revolution (T) using the formula T = 2πr / v. However, it seems like there might be an error in your calculation of r.
You mentioned that r = 0.83tan21, but I'm afraid that's not quite right. Since the length of the string is given as 83.0 cm, we need to convert it to meters by dividing by 100. So, r = (0.83 / 100)tan(21).
Let's recalculate that value: r = 0.83/100 * tan(21) ≈ 0.0297 m.
Now we can calculate the time it takes to make one revolution: T = 2πr / v.
Substituting the values: T = 2π * 0.0297 / 1.094786688 ≈ 0.543 seconds.
So, it takes approximately 0.543 seconds to make one full revolution.
Now, let's move on to the tension in the string. You correctly calculated that the vertical component of the tension (Ty) is equal to the weight of the mass: Ty = 9 * 9.8 = 88.2 N.
However, the horizontal component of the tension (Tx) is not equal to 9a. Remember that the net force in the x-direction is zero since the mass moves in a circle at constant speed. Therefore, Tx = 0.
Since the tension in the string is the vector sum of Tx and Ty, we can use the Pythagorean theorem to find the tension (T):
T² = Tx² + Ty²
T² = 0² + 88.2²
T ≈ 88.2 N.
So, the tension in the string is approximately 88.2 N.
I hope that helps solve the mystery! If you have any more questions, feel free to ask!
To correct your calculation for the time it takes to make one revolution, you need to use the correct formula. The formula you used, T = 2πr/v, gives the period of the circular motion, not the time for one revolution.
The correct formula to use is T = 2πr/v = 2π(r/v).
Given the values you have calculated so far:
r = 0.83 tan(21°) = 0.305 m
v = √(a * r) = √((88.2 tan(21°) / 9) * 0.305) ≈ 1.095 m/s
Now you can calculate the time for one revolution:
T = 2π(0.305 m) / 1.095 m/s ≈ 1.77 s
So it takes approximately 1.77 seconds to make one revolution.
For the tension in the string, you have correctly calculated:
T = (88.2 tan(21°)) / sin(21°) ≈ 94.8 N
So the tension in the string is approximately 94.8 N.
To find how long it takes to make one revolution, we can use the relationship between velocity, radius, and period.
1. We can start by calculating the radius of the circular motion. The string length is given as L = 83.0 cm, and the angle with the vertical is given as theta = 21 degrees. We can use the tangent function to find the vertical component of the length:
r = L * tan(theta)
= 0.83 m * tan(21)
≈ 0.307 m
2. The centripetal acceleration a can be found by considering the forces acting on the mass. The tension in the string provides the centripetal force, so we can set up an equation for the sum of forces in the vertical direction:
Ty - mg = 0
Solving for Ty, we get:
Ty = mg
= 9.0 kg * 9.8 m/s^2
≈ 88.2 N
3. The centripetal acceleration a can be calculated by considering the horizontal component of the tension force:
Fx = Tx = ma
Solving for a:
a = Tx / m
= 88.2 N * tan(theta) / 9.0 kg
≈ 22.74 m/s^2
4. The velocity v of a point moving in a circle at constant speed is related to the centripetal acceleration and radius by the equation:
v = sqrt(a * r)
Substituting the values we have:
v = sqrt(22.74 m/s^2 * 0.307 m)
≈ 1.089 m/s
5. Finally, we can find the time period T for one revolution by using the relation:
T = 2 * pi * r / v
Substituting the values we have:
T = 2 * pi * 0.307 m / 1.089 m/s
≈ 1.778 s
Therefore, it takes approximately 1.778 seconds to make one revolution.
Regarding the tension in the string, based on your calculations, it seems you have already obtained the correct answer. The tension in the string is approximately 94.8 N.