One end of a cord is fixed and a small 0.270-kg object is attached to the other

Question

One end of a cord is fixed and a small 0.270-kg object is attached to the other end, where it swings in a section of a vertical circle of radius 3.00 m as shown in the figure below. When = 16.0°, the speed of the object is 6.30 m/s. At this instant, find each of the following.

(a) the tension in the string
T =  N

(b) the tangential and radial components of acceleration
ar =  m/s2 inward
at =  m/s2 downward tangent to the circle

(c) the total acceleration
atotal =  m/s2 inward and below the cord at  °

(d) Is your answer changed if the object is swinging down toward its lowest point instead of swinging up?

Yes or No     

(e) Explain your answer to part (d).

Explanation / Answer

A mass of object 0.270 kg

vertical circle of radius 3 m .

the angle is 16,

the speed of the object is 6.3 m/s.

The tension in the supports the weight of the object and provides the centripetal force that keeps the object moving in a circular path.

The vertical component of tension = T * cos 16
The vertical component of the tension supports the weight of the object.
T * cos 16 = m * g = 0.27 * 9.8 =2.64
T = 2.64 ÷ cos 16 =2.740 N

This is the tension if the object was not moving.

Centripetal force = m * v2/r = 0.27 * 6.32 /3 = 3.57 N
Total tension in cord =2.740 + 3.57= 6.31 N
The 6.31 N tension force is directed toward the center of the circle. So, all of the tension is the force causing the radial acceleration.
The component of the weight that is directed toward the center = 0.27 * 9.8 * cos 16

The net force that is directed toward the center = 6.31 – 0.27 * 9.8 * cos 16
Radial acceleration = net radial force ÷ mass
Radial acceleration = 6.31 – 0.27 * 9.8 * cos 16÷ 0.27 = 13.95 m/s2


The direction of the tangential acceleration is perpendicular to the tension force in the cord, so the tension has no component causing tangential acceleration. The weight has a component that is tangent to the circle.
The tangential force = m* g * sin = 0.27 * 9.8 * sin 16=0.73
Tangential acceleration = tangential force ÷ mass = 2.70 m/s2

Total acceleration = (2.702 + 13.952)0.5 = 14.20 m/s2

d) Is your answer changed if the object is swinging down toward its lowest point instead of swinging up? NO
The weight and centripetal force are not dependent on the direction the object is moving. So the radial and tangential acceleration are not dependent on the direction the object is moving.

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