A conical pendulum is formed by attaching a 0.60kg ball to a 1.0m -long string, then allowing the mass to move in a horizontal circle of radius 30cm.
1. What is the tension in the string
2. What is the ball's angular velocity in rpm?
sinα=R/L=0.3/1=0.3
α=17.45°
mv²/R=Tsinα …..(1)
mg=Tcosα …..(2)
Divide (1) by (2)
v²/Rg=tanα
v=sqrt(R•g•tanα)=
=sqrt(0.3•9.8•0.31)=0.27 m/s
ω=v/R=0.27/0.3=0.9 rad/s
f= ω/2π=0.9/2•3.14=0.14rev/s=8.6 rev/min
T=mg/cosα=0.6•9.8/0.95=6.16 N
To find the tension in the string, you can use the concept of centripetal force. In a conical pendulum, the centripetal force is provided by the tension in the string.
1. Tension in the string:
The centripetal force can be found using the formula:
F = M * omega^2 * R
Where:
F is the centripetal force
M is the mass of the ball (0.60kg in this case)
omega is the angular velocity (which we'll solve for in the second question)
R is the radius of the circle (30cm or 0.30m in this case)
Now, since the centripetal force is provided by the tension in the string, the tension will be equal to the centripetal force. So:
Tension = F = M * omega^2 * R = 0.60kg * omega^2 * 0.30m
2. Angular velocity in rpm:
To find the angular velocity, you can use the formula:
omega = v / R
Where:
omega is the angular velocity
v is the linear velocity of the ball
The linear velocity v can be found using the formula:
v = 2 * pi * R * n
Where:
v is the linear velocity
R is the radius of the circle (same as before) and
n is the number of revolutions per minute (rpm) that the ball makes.
Now, substituting the value of linear velocity v into the formula for angular velocity omega, we can solve for omega in terms of rpm:
omega = 2 * pi * R * n / R = 2 * pi * n
Now you can substitute the given values for R and n to find the angular velocity in rpm:
omega = 2 * pi * n = 2 * pi * 1 / 60 = 0.105 radians per second
To convert this into rpm, you can use the conversion factor:
1 revolution = 2 * pi radians
So,
0.105 radians per second * (1 revolution / 2 * pi radians) * 60 seconds = 3.98 rpm
Therefore, the ball's angular velocity is approximately 3.98 rpm.