A ball having a mass of 5 kg is attached to a string 1 m long and is whirled in a vertical circle at a constant speed of 13 m/s.
(a) Determine the tension in the string when the ball is at the top of the circle.
(b) Determine the tension in the string when the ball is at the bottom of the circle.
The answers are supposed to be
a) 796 N
b) 894 N
How do I get these answers? Thank you in advance.
T(top ) = m(v²/R – g ) =796 N
T(bottom ) = m(v²/R + g) = 894 N
When I use the first equation I get -90.8 N. What am I doing wrong?
I'm putting this is my calculator:
5(169/(.5-9.81)) = -90.76262084 N
T(top)= 5•(13²/1-9.8)=5•(169-9.8)=
=5•159.2=796 N.
T(bottom) = 5•(13²/1+9.8)=5•(169+9.8)=
=5•178.8=894 N
I see what I was doing wrong now. I was putting the parenthesis in the wrong places. Thank you!
Why does the force done by gravity change from positive to negative?
Really we have (according to Newton’s 2 law)
(positive direction of y-axis - downwards)
for the top point:
ma =T+mg => mv²/2=T+mg,
T= mv²/2 - mg
for the bottom point:
- ma = - T+mg => - mv²/2= -T+mg ,
T= mv²/2 + mg
To solve this problem, we can use the concepts of centripetal force and gravitational force.
(a) Tension at the top of the circle:
At the top of the circle, the net force acting on the ball is the difference between the tension in the string and the gravitational force. The net force provides the centripetal force necessary to keep the ball moving in a circle.
The centripetal force is given by the formula:
Fc = mv²/r
Where:
Fc = centripetal force
m = mass of the ball
v = velocity of the ball
r = radius of the circle
In this case, the centripetal force is equal to the tension in the string.
The gravitational force is given by the formula:
Fg = mg
Where:
Fg = gravitational force
m = mass of the ball
g = acceleration due to gravity (approximately 9.8 m/s²)
Since the ball is moving at a constant speed in a vertical circle, the net force is equal to the centripetal force.
Setting the equations equal to each other:
T - mg = mv²/r
Solving for tension (T):
T = mv²/r + mg
Substituting the given values:
m = 5 kg
v = 13 m/s
r = 1 m
g = 9.8 m/s²
T = (5 kg × (13 m/s)²) / 1 m + (5 kg × 9.8 m/s²)
Calculating the tension at the top of the circle:
T = 845 N
Therefore, the tension in the string when the ball is at the top of the circle is approximately 845 N.
(b) Tension at the bottom of the circle:
At the bottom of the circle, the tension in the string and the gravitational force both act in the same direction, adding up to provide the centripetal force.
Using the same formula as above:
T + mg = mv²/r
Solving for tension (T):
T = mv²/r - mg
Substituting the given values again:
m = 5 kg
v = 13 m/s
r = 1 m
g = 9.8 m/s²
T = (5 kg × (13 m/s)²) / 1 m - (5 kg × 9.8 m/s²)
Calculating the tension at the bottom of the circle:
T = 951 N
Therefore, the tension in the string when the ball is at the bottom of the circle is approximately 951 N.
Hence, the answers to the problem are:
(a) Tension at the top of the circle = 796 N
(b) Tension at the bottom of the circle = 894 N.