An object attached to the end of a string swings in a vertical circle (R = 1.2 m), as shown. At an instant when ƒá = 30„a, the speed of the object is 5.1 m/s and the tension in the string has a magnitude of 20 N. What is the mass of the object?

To find the mass of the object, we can use the concepts of circular motion and centripetal force.

Let's start by defining the variables:

R = radius of the circular path = 1.2 m
ƒá = angle made by the string with the vertical direction = 30°
v = speed of the object = 5.1 m/s
T = tension in the string = 20 N
m = mass of the object (what we need to find)

The centripetal force, which keeps the object moving in a circular path, is provided by the tension in the string. The tension force and the gravitational force are the only forces acting on the object in the vertical direction.

At the moment when ƒá = 30°, we can use trigonometry to find the vertical component of the tension force (T_y) using the angle ƒá:

T_y = T * sin(ć)

The vertical component of the tension force (T_y) is equal to the gravitational force (mg), where g is the acceleration due to gravity.

T_y = mg

Therefore, we can equate these two equations to find the mass of the object:

mg = T * sin(ć)

Now, let's substitute the known values into the equation:

m * g = 20 * sin(30°)

To solve for m, we need the value of g, which is approximately 9.8 m/s².

m * 9.8 = 20 * sin(30°)

Now, let's calculate the value of sin(30°) and solve for m:

sin(30°) = 0.5

m * 9.8 = 20 * 0.5

m * 9.8 = 10

Divide both sides of the equation by 9.8 to isolate the mass (m):

m = 10 / 9.8

m ≈ 1.02 kg

Therefore, the mass of the object is approximately 1.02 kg.

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