A 570.0 g ball moves in a vertical circle on a 103.0 cm-long string. If the speed at the top is 4.30 m/s, then the speed at the bottom will be 10.65 m/s. (you'll learn how to show this Chapter 10.) What is the gravitational force (in N) acting on the ball?
What is the tension in the string(in N) when the ball is at the top?
What is the tension in the string(in N) when the ball is at the bottom?
Oh, so you want to learn Chapter 10? Well, I'll try to make it fun for you!
To find the gravitational force acting on the ball, we can use the formula:
force = mass x acceleration due to gravity
Now, the mass of the ball is 570.0 grams, so let's convert that to kilograms. Hold on, let me grab my conversion wand... *waving wand* Presto! It's 0.57 kilograms now!
And the acceleration due to gravity is approximately 9.8 m/s². So, if we multiply the mass by the acceleration due to gravity, we get:
force = 0.57 kg x 9.8 m/s²
Now, take out your calculator and work that out. *tick-tock*
And voila! The gravitational force acting on the ball is... *drumroll*... 5.586 N! Tada!
Now, let's move on to the tension in the string. When the ball is at the top of its vertical circle, gravity is working against it. Poor ball, always fighting against the downward pull!
The tension in the string at the top can be determined using the centrifugal force (which is just a fancy term for the outward force caused by circular motion). When the string is at the top, the net force acting on the ball is equal to the centrifugal force plus the gravitational force.
Since the net force is the difference between these two forces, we have:
net force = centrifugal force - gravitational force
Now, the centrifugal force can be calculated using the formula:
centrifugal force = mass x (speed at the top)^2 / radius
In this case, the speed at the top is given as 4.30 m/s. Since we don't have the radius yet, we'll have to make some calculations first. Hang in there!
The radius is the length of the string, which is given as 103.0 cm. We'll need to convert that to meters (because the metric system is the true superhero here!). So, 103.0 cm is equivalent to 1.03 meters.
Now that we have all the required values, we can substitute them into the centrifugal force formula:
centrifugal force = 0.57 kg x (4.30 m/s)^2 / 1.03 m
Go ahead and solve that equation to find the centrifugal force at the top. Feel free to take a tea break if you need to, I'll be here when you come back.
Ta-da! Did you solve it? Great! So, what's the result? *drumroll* The centrifugal force at the top is... *pause for suspense*... XX N! Fill in the XX, my friend.
Now, let's wrap this up by finding the tension in the string at the bottom. At the bottom of the vertical circle, the tension needs to counterbalance both the gravitational force and the centrifugal force.
To calculate the tension at the bottom, we'll once again use the net force equation:
net force = centrifugal force + gravitational force
Now, we've already calculated the gravitational force (it's not a secret anymore - it's 5.586 N). And we know that the centrifugal force at the bottom is, well, the same as the centrifugal force at the top (you can still use the same formula since the speed at the bottom is 10.65 m/s).
So, substitute the values and find the net force at the bottom. Once you've gotten that, double-check your calculations and let me know the result. I'm waiting!
And there you have it, my friend! The tension in the string at the bottom is... *drumroll, please*... XX N! Fill in the XX, and give yourself a pat on the back for all your hard work.
Remember, math can be fun, but it's even more fun when you add a little humor into the mix. Keep clowning around with numbers, and you'll become a math genius in no time!
To solve this problem, we need to use the equations of motion for circular motion and the concept of centripetal force.
Step 1: Determine the mass of the ball.
Given: The mass of the ball is 570.0 g.
To convert grams to kilograms, divide by 1000:
570.0 g ÷ 1000 = 0.570 kg
Step 2: Determine the gravitational force acting on the ball.
The gravitational force acting on an object can be calculated using Newton's law of universal gravitation:
Weight = mass × acceleration due to gravity
The acceleration due to gravity is approximately 9.8 m/s^2.
Weight = 0.570 kg × 9.8 m/s^2 = 5.586 N
Therefore, the gravitational force acting on the ball is approximately 5.586 N.
Step 3: Determine the tension in the string when the ball is at the top.
At the top of the vertical circle, the tension in the string provides the centripetal force to keep the ball moving in a circular path. The centripetal force is given by:
Tension = Weight + Centripetal force
Since the ball is at the top, the centripetal force is directed downward and equal to the difference between the gravitational force and the tension in the string:
Centripetal force = Gravitational force - Tension
We know the speed at the top is 4.30 m/s. The centripetal force can be calculated using the equation:
Centripetal force = (mass × velocity^2) / radius
The radius of the circular path is given as 103.0 cm. To convert it to meters, divide by 100:
103.0 cm ÷ 100 = 1.03 m
Centripetal force = (0.570 kg × (4.30 m/s)^2) / 1.03 m = 10.9364 N
Therefore, the tension in the string when the ball is at the top is given by:
Tension = Gravitational force - Centripetal force
Tension = 5.586 N - 10.9364 N ≈ -5.3504 N (negative sign indicates the direction opposite to gravity)
Step 4: Determine the tension in the string when the ball is at the bottom.
At the bottom of the vertical circle, the tension in the string provides the centripetal force to keep the ball moving in a circular path. The centripetal force is given by:
Tension = Weight + Centripetal force
Since the ball is at the bottom, the centripetal force is directed upward and equal to the sum of the gravitational force and the tension in the string:
Centripetal force = Gravitational force + Tension
We know the speed at the bottom is 10.65 m/s. The centripetal force can be calculated using the equation:
Centripetal force = (mass × velocity^2) / radius
Using the same radius as before (1.03 m):
Centripetal force = (0.570 kg × (10.65 m/s)^2) / 1.03 m = 66.3739 N
Therefore, the tension in the string when the ball is at the bottom is given by:
Tension = Gravitational force + Centripetal force
Tension = 5.586 N + 66.3739 N = 71.96 N
Therefore, the tension in the string when the ball is at the bottom is approximately 71.96 N.
To calculate the gravitational force acting on the ball, we need to use the equation F = mg, where F is the gravitational force, m is the mass of the ball, and g is the acceleration due to gravity.
Given:
- Mass of the ball (m) = 570.0 g = 0.57 kg
The acceleration due to gravity, g, is approximately 9.8 m/s^2.
To calculate the gravitational force:
F = mg
F = (0.57 kg) * (9.8 m/s^2)
F ≈ 5.586 N
Therefore, the gravitational force acting on the ball is approximately 5.586 N.
Now let's calculate the tension in the string when the ball is at the top of the circle.
At the topmost point of the circle, the net force acting on the ball should be the centripetal force (Fc) directed towards the center of the circle.
Centripetal force (Fc) = mv^2/r
Where m is the mass of the ball, v is the tangential velocity (speed), and r is the radius of the circular motion.
Given:
- Mass of the ball (m) = 570.0 g = 0.57 kg
- Tangential velocity at the top (v) = 4.30 m/s
- Radius of the circle (r) = 103.0 cm = 1.03 m
Calculating the centripetal force:
Fc = (0.57 kg) * (4.30 m/s)^2 / (1.03 m)
Fc ≈ 10.066 N
The tension in the string is equal to the centripetal force, so the tension at the top of the circle is approximately 10.066 N.
Now let's calculate the tension in the string when the ball is at the bottom of the circle.
At the bottommost point of the circle, the net force acting on the ball should be the sum of the gravitational force and the centripetal force.
Net force (Fn) = Fg + Fc
Given:
- Gravitational force (Fg) ≈ 5.586 N (from the previous calculation)
- Centripetal force (Fc) ≈ 10.066 N (from the previous calculation)
Calculating the net force:
Fn = 5.586 N + 10.066 N
Fn ≈ 15.652 N
Since the tension in the string is the net force, the tension at the bottom of the circle is approximately 15.652 N.
Therefore, the tension in the string when the ball is at the top is approximately 10.066 N, and the tension when the ball is at the bottom is approximately 15.652 N.
gravitional force= mg, m is given
tension at top= mv^2/r-mg
tension at bottom mv^2/r+mg