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.
Sorry something went wrong with my internet and this was posted twice.
To find the tension in the string at the top and bottom of the circle, we can start by analyzing the forces acting on the ball at each position.
Let's begin with the top of the circle.
(a) Tension at the top of the circle:
At the top of the circle, the ball is moving in a circular path, so the net force acting on it must provide the centripetal force necessary for this motion.
1. Start by examining the forces acting on the ball at the top:
- The weight of the ball (mg), which acts vertically downwards.
- The tension in the string, which acts towards the center of the circle (towards the bottom of the ball).
2. To determine the tension, let's use the net force equation:
Net Force = (mass x velocity^2) / radius
In this case, the velocity is given as 13 m/s, and the radius of the circle is 1 m.
Plugging in these values, we have:
Tension - mg = (m x v^2) / r
3. Solve for tension:
Tension = (m x v^2) / r + mg
Plugging in the values: mass (m) = 5 kg, velocity (v) = 13 m/s, and radius (r) = 1 m,
Tension = (5 x 13^2) / 1 + (5 x 9.8)
= (5 x 169) / 1 + (5 x 9.8)
= 845 / 1 + 49
= 845 / 50
= 16.9 + 9.8
= 26.7 m/s
Therefore, the tension in the string at the top of the circle is approximately 26.7 N.
Moving on to the bottom of the circle:
(b) Tension at the bottom of the circle:
At the bottom of the circle, the ball is still moving in a circular path, but now the net force acting on it must provide both the centripetal force and extra force to counterbalance the weight.
1. Start by examining the forces acting on the ball at the bottom:
- The weight of the ball (mg), which acts vertically downwards.
- The tension in the string, which acts towards the center of the circle (towards the top of the ball).
2. Using the net force equation again:
Tension + mg = (mass x velocity^2) / radius
3. Solve for tension:
Tension = (m x v^2) / r - mg
Plugging in the values: mass (m) = 5 kg, velocity (v) = 13 m/s, and radius (r) = 1 m,
Tension = (5 x 13^2) / 1 - (5 x 9.8)
= (5 x 169) / 1 - (5 x 9.8)
= 845 / 1 - 49
= 845 / 50
= 16.9 - 9.8
= 7.1 m/s
Therefore, the tension in the string at the bottom of the circle is approximately 7.1 N.
So the correct answers are:
(a) The tension in the string at the top of the circle is approximately 26.7 N.
(b) The tension in the string at the bottom of the circle is approximately 7.1 N.
To solve this problem, you need to consider the forces acting on the ball at different positions in the vertical circle: the force of gravity (weight) and the tension in the string.
(a) When the ball is at the top of the loop, the tension in the string and the weight are the two forces acting on the ball. The net force provides the centripetal force necessary for the circular motion.
The formula for net force in circular motion is:
Net force = Mass × (Centripetal acceleration)
At the top of the circle, the net force is equal to the difference between the tension and the weight (since they are in opposite directions):
Tension - Weight = Mass × (Centripetal acceleration)
To find the value of the tension, substitute the given values into the equation:
Tension - (Mass × gravity) = Mass × (velocity^2 / radius)
Mass = 5 kg
Gravity = 9.8 m/s^2
Velocity = 13 m/s
Radius = 1 m
Tension - (5 kg × 9.8 m/s^2) = 5 kg × (13 m/s)^2 / 1 m
Solving for the tension gives:
Tension = 5 kg × (13 m/s)^2 / 1 m + 5 kg × 9.8 m/s^2
= 845 N + 49 N
= 894 N
Therefore, the tension in the string when the ball is at the top of the circle is 894 N.
(b) When the ball is at the bottom of the circle, the net force is equal to the sum of the tension and the weight (since they are in the same direction):
Tension + Weight = Mass × (Centripetal acceleration)
Using the same formula and substituting the given values:
Tension + (5 kg × 9.8 m/s^2) = 5 kg × (13 m/s)^2 / 1 m
Solving for the tension gives:
Tension = 5 kg × (13 m/s)^2 / 1 m - 5 kg × 9.8 m/s^2
= 845 N - 49 N
= 796 N
Therefore, the tension in the string when the ball is at the bottom of the circle is 796 N.
By using the formulas for net force and the given information about the ball's mass, speed, and radius, you can calculate the tensions at the top and bottom of the vertical circle.