One end of a cord is fixed and a small 0.560-kg object is attached to the other

Question

One end of a cord is fixed and a small 0.560-kg object is attached to the other end, where it swings in a section of a vertical circle of radius 1.50 m, as shown in the figure below. When = 27.0°, the speed of the object is 7.90 m/s.

(a) At this instant, find the tension in the string.
T = ___N

(b) At this instant, find the tangential and radial components of acceleration.

(c) At this instant, find 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

The vertical component of tension = T * cos 27
The vertical component of the tension supports the weight of the object.
T * cos 27 = m * g = 0.56 * 9.8 = 5.48
T = 5.48 ÷ cos 27 = 6.16 N This is the tension if the object was not moving.

Centripetal force = m * v^2/r = 0.56 * 7.9^2 /1.5 = 23.3 N
Total tension in cord = 5.48 + 23.3 = 28.78 N
The 28.78 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 = 0.56 * 9.8 * cos 27

The net force that is directed toward the center = 28.78 – 0.56 * 9.8 * cos 27
Radial acceleration = net radial force ÷ mass
Radial acceleration = (28.78 – 0.56 * 9.8 * cos 27) ÷ 0.56 = 42.6 m/s^2


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 = 0.56 * 9.8 * sin 27
Tangential acceleration = tangential force ÷ mass = 9.8 * sin 27 = 4.45 m/s^2

Total acceleration = (42.6^2 + 4.45^2)^0.5 = 42.8 m/s^2

c)

no

d)

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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