5. Consider a ball on a fixed-length string being whirled in a vertical circular

Question

5. Consider a ball on a fixed-length string being whirled in a vertical circular path as shown in the diagram below.

(a) When the ball is at the top of the circle, what is the expression for the centripetal force on the ball? Take the upward direction as positive and downward direction as negative when considering the sign of the forces. (Use the following as necessary: m,g, and T.)

Fc=_______________

(b) If the speed of the ball at the top of the circle is v what is the expression for the centripetal force on the ball in terms of its speed v and radius L? Take the upward direction as positive and downward direction as negative when considering the sign of the forces. (Use the following as necessary: m, v, and L.)

Fc=_______________

(c) Use your answers in parts (d) and (e) to get an expression for the tension in the string in terms of the speed of the ball and the length of the string. (Use the following as necessary: m, v, L, and g.)

T=________________

(d) What minimum speed should the ball have at the top of the circle to continue in its circular motion? (Use the following as necessary: m, v, L, and g.)

vmin=______________

Explanation / Answer

(a) When the ball is at the top of the circle, what is the expression for the centripetal force on the ball?

from newtons second law

Fc=marad .

arad = v^2/L

Fc=mv^2/L

(b) If the speed of the ball at the top of the circle is v what is the expression for the centripetal force on the ball in terms of its speed v and radius L?

Fc=mv^2/L

(c) Use your answers in parts (d) and (e) to get an expression for the tension in the string in terms of the speed of the ball and the length of the string

Top of the circle:

Let's calculate the tension in the string at the top of the circle. Both of the forces T and mg are directed downwards towards the center. Since our block is in circular motion, we know that the NET FORCE must act towards the center of the circle.

net force to the center = T + mg

Fc = T + mg
m(v2/L) = T + mg
T = m(v2/L) - mg

(d) What minimum speed should the ball have at the top of the circle to continue in its circular motion?

The minimum or critical velocity needed for the block to just be able to pass through the top of the circle without the rope sagging then we would start by letting the tension in the rope approaches zero.

0 = m(v2/L) - mg

m(v2/L) = mg
v2/L = g
v2 = Lg
vmin = (Lg)

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