ball of mass m = 0.2 kg is attached to a (massless) string of length L = 3 m and is undergoing circular motion in the horizontal plane, as shown in the figure.
What should the speed of the mass be for θ to be 46°?
What is the tension in the string?
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To find the speed of the mass for a given angle θ, we can use the concept of circular motion and the centripetal force.
We know that the tension in the string provides the centripetal force required for the circular motion. The centripetal force is given by the equation:
Fc = m * v^2 / R
where Fc is the centripetal force, m is the mass of the ball, v is the speed of the ball, and R is the radius of the circular path.
In this case, the radius of the circular path is given by the length of the string, L. So R = L = 3 m.
We are given the mass of the ball, m = 0.2 kg, and the angle θ = 46°.
To find the speed of the mass, we need to find the tension in the string, which is equal to the centripetal force. Rearranging the formula for centripetal force, we get:
Fc = Tension = m * v^2 / R
Substituting the given values, we have:
Tension = (0.2 kg) * v^2 / 3 m
Now, we need to find the tension in the string. To do this, we need to find the value of v.
We can relate the angle θ and the length L of the string using trigonometry:
sin(θ) = opposite / hypotenuse = L / R = L / 3
Substituting the given value of L = 3 m and θ = 46°, we have:
sin(46°) = 3 / 3
Simplifying this equation, we get:
sin(46°) = 1
Now, we can find the value of v by rearranging the trigonometric equation:
v = L * sin(θ)
Substituting the values of L = 3 m and θ = 46°, we have:
v = 3 m * sin(46°)
Using a calculator, we can find the value of sin(46°) which is approximately 0.7193.
Substituting this value, we have:
v = 3 m * 0.7193
Simplifying, we get:
v ≈ 2.1579 m/s
Therefore, the speed of the mass should be approximately 2.16 m/s for θ to be 46°.
Now, to find the tension in the string, we substitute the value of velocity into the equation:
Tension = (0.2 kg) * (2.1579 m/s)^2 / 3 m
Using a calculator to evaluate this equation, we get:
Tension ≈ 0.309 N
Therefore, the tension in the string is approximately 0.309 N.