Centripetal force Objecties Describe and analyze the motion of objects moing in circular motion. Apply Newton s second law to circular motion problems. Interpret free-body force diagrams. 1. A race car is moing with a speed of 200 km/h on a circular section of a race track that has a radius of 300 m. The race car and drier hae a mass of 800 kg. a. What is the magnitude of the centripetal acceleration felt by the drier? 2. In the case of a car rounding a turn, what real physical force is acting radially to keep the car moing in a circular path? b. What is the centripetal force acting on the mass? Physics terms Equations centripetal force centripetal acceleration 1
Consider Newton s first law An object in motion remains in motion in a straight line at constant speed unless acted upon by an unbalanced force. Circular motion Imagine whirling a yo-yo around your head in a circle. Do you hae to exert a force on the yo-yo? Circular motion Imagine whirling a yo-yo around your head in a circle. Do you hae to exert a force on the yo-yo? Yes! Which way does this force ector point? Inestigation In Inestigation 7A you will examine the elocity, acceleration, and force ectors for an object in circular motion. Click this simulation on page 210. Inestigation Part 1: Directions of the ectors 1. Set m = 5.0 kg, r = 5.0 m, and = 5.0 m/s. 2. Play the simulation and pause it at arious positions around the circle. 3. Sketch the elocity, force, and acceleration ectors for at least fie positions distributed around the circle. Inestigation Questions for Part 1 a. Which ector quantities are radial and which are tangential? Are the radial quantities pointed into the center or outwards? b. Do the lengths of the elocity, acceleration, or force ectors change around the circle? c. Notice that the angular elocity is exactly 1 rad/sec. Why? 2
Inestigation Part 2: Approximating a mass swung oerhead 1. Set r = 1.0 m and m = 0.3 kg 2. Calculate the tangential elocity needed to spin the object once per second, and enter into simulation. Inestigation Questions for Part 2 a. How much force is needed to maintain the object in circular motion? b. Compare this with the force required to hold the object motionless against the force of graity. c. Lengthen the string to r = 2.0 m. Do you need more or less force to maintain the same angular elocity? Inestigation Part 3: Variation of elocity with radius 1. Hold the force constant at 10 N and the mass constant at 2 kg, but ary the length of the string from r = 1 m to 5 m. 2. Record the elocity and radius for each case. Inestigation Questions for Part 3 a. Graph (on the ertical axis) against r and describe the shape of your graph. b. Graph 2 against r, describe the shape of your graph, and measure its slope (including units). Is it accelerating? This object is moing at constant speed. Is it accelerating? Acceleration at constant speed This object is moing at constant speed. Is it accelerating? Hint: Is the elocity ector changing? YES! It is accelerating. The elocity ector is constantly changing direction. 3
Acceleration at constant speed This object is moing at constant speed. Is it accelerating?... constantly redirects the elocity ector. always points toward the center of the circle. Uniform circular motion is the weird case of acceleration at constant speed. a c a c a c a c How can you calculate a c? What is the mathematical relationship between elocity and centripetal acceleration? How can you calculate a c? What is the mathematical relationship between elocity and centripetal acceleration? a c a c a c a c Where does this equation come from? Deriation: is: An object in circular motion moes from point A to point B at constant speed. The distance from A to B is d = Δt. The change in elocity is Δ. The blue and gray triangles are similar. Use this to get an expression for Δ. 4
Similar triangles mean that: Similar triangles mean that: Therefore, centripetal acceleration is: Therefore, centripetal acceleration is: Notice: Δ always points toward the center of the circle. Exploring the ideas Click the interactie calculator on page 211. What is a c for a mass spun around on a 1.0-m-long string if the mass completes one rotation eery second? Hint: First, find the elocity. 1.0 What is a c for a mass spun around on a 1.0-m-long string if the mass completes one rotation eery second? 6.28 What is a c for a mass spun around on a 1.0-m-long string if the mass completes one rotation eery second? 6.28 Click to iew motion 39.5 1.0 1.0 Now find the acceleration. Now find the acceleration. This is about 4 g s of acceleration! It s not unusual for a c to be large. 5
It takes a force to cause an object to moe in a circle. This force is called the centripetal force. Any type of physical force can proide a centripetal force. What supplies the centripetal force in this situation? F c Any type of physical force can proide a centripetal force. What supplies the centripetal force in this situation? F c = T The centripetal force is the force or combination of forces that point toward the center of the circle. F c = T The string supplies the centripetal force. The centripetal force is the force of tension. T T What if the string breaks? Which way will the yo-yo go? What if the string breaks? Which way will the yo-yo go? It flies off in a straight line tangent to the circle. It can t stay in the circle unless a centripetal force is applied. 6
Newton s second law A centripetal force causes an object to undergo centripetal acceleration. Combining these two equations... and The centripetal force and acceleration ectors must point in the SAME direction: toward the center of the circle. F c a c Exploring the ideas Combining these two equations... you get: and Click the interactie calculator on page 211. A 200 gram yo-yo on a 1.0 meter string is spinning in a circle once eery second. What is the tension in the string? 0.200 A 200 gram yo-yo on a 1.0 meter string is spinning in a circle once eery second. What is the tension in the string? 0.200 6.28 Hint: First find the elocity. 1.0 1.0 Next, find the centripetal force. 7
A 200 gram yo-yo on a 1.0 meter string is spinning in a circle once eery second. What is the tension in the string? 0.200 6.28 A 1000 kg car traeling at 10 m/s is rounding a cure with a radius of 50 m. How much force is needed to keep it on the road? 1000 10 7.9 1.0 50 A 1000 kg car traeling at 10 m/s is rounding a cure with a radius of 50 m. How much force is needed to keep it on the road? 1000 10 A 1000 kg car traeling at 10 m/s is rounding a cure with a radius of 50 m. How much force is needed to keep it on the road? 1000 20 2000 8000 50 50 If the speed doubles, how much more force is needed? If the speed doubles, how much more force is needed? four times more (8000 N) Centrifugal s. centripetal force The terms centripetal and centrifugal sound so similar. What s the difference? Centripetal means center-seeking. A centripetal force pushes inward, toward the center of the circle. Centrifugal means center-fleeing. A centrifugal force pushes outward, away from the circle. One of these forces is referred to as fictitious. Which one, and why? Centrifugal force Imagine that you are a small child in a car making a sharp turn. As the car turns, you slide sideways across the seat until you are pressed against the far wall of the car (unless you wear a seatbelt! ). It FEELS like a force is pushing you OUT of the circle. Why? 8
Centrifugal force It FEELS like a force is pushing you OUT of the circle. Why? You are obeying Newton s first law. You are moing in a straight line UNLESS acted upon by a force. Centrifugal force It FEELS like a force is pushing you OUT of the circle. Why? You are obeying Newton s first law. You are moing in a straight line UNLESS acted upon by a force. There is NO ACTUAL FORCE pushing you out of the circle. That is why say the centrifugal force is called fictitious. Centrifugal force Centrifugal force Once you reach the end of the seat, the car door will push on you to keep you in the circle. This ACTUAL force is the centripetal force. F c F c The centrifugal force proides a sensation that feels ery real. Be careful when soling circular motion problems that you are not tricked into including this fictitious force on free-body diagrams. acceleration in a turn is 1 m/s 2. What is the minimum radius cure a ciil engineer should design on a road where cars trael at 30 m/s (67 mph)? acceleration in a turn is 1 m/s 2. What is the minimum radius cure a ciil engineer should design on a road where cars trael at 30 m/s (67 mph)? Asked: r Gien: a Relationships: 9
acceleration in a turn is 1 m/s 2. What is the minimum radius cure a ciil engineer should design on a road where cars trael at 30 m/s (67 mph)? Asked: r Gien: a c, Relationships: acceleration in a turn is 1 m/s 2. What is the minimum radius cure a ciil engineer should design on a road where cars trael at 30 m/s (67 mph)? Asked: r Gien: a c, Relationships: acceleration in a turn is 1 m/s 2. How much force is needed to create this acceleration for a 1200 kg car? acceleration in a turn is 1 m/s 2. How much force is needed to create this acceleration for a 1200 kg car? What is proiding this force? static friction What might happen if the road is icy? acceleration in a turn is 1 m/s 2. How much force is needed to create this acceleration for a 1200 kg car? The free-body diagram for a car safely rounding a cure shows the friction, which proides the centripetal force. What is proiding this force? static friction What might happen if the road is icy? F f SIDE VIEW If there is less than 1200 N of friction aailable, the car will slide off the road, tangent to the circle. F N mg F N VIEW FROM ABOVE 10
The friction must point toward the center of the circle. What does high-g mean? This airplane s high elocity as it moes through a tight turn results in a centripetal acceleration of 100 m/s 2. This is about 10 times the acceleration of graity (g = 9.8 m/s 2 ). F f center of circle F N mg F N What does high-g mean? This airplane s high elocity as it moes through a tight turn results in a centripetal acceleration of 100 m/s 2. This is about 10 times the acceleration of graity (g = 9.8 m/s 2 ). 1. A race car is moing with a speed of 200 km/h on a circular section of a race track that has a radius of 400 m. The race car and drier hae a mass of 1400 kg. a) What is the magnitude of the centripetal acceleration felt by the drier? The centripetal force needed on a 70 kg pilot would be 7000 N. At this acceleration, the heart cannot pump blood to the brain and the pilot would lose consciousness. Asked: a Gien: Relationship: 1. A race car is moing with a speed of 200 km/h on a circular section of a race track that has a radius of 400 m. The race car and drier hae a mass of 1400 kg. a) What is the magnitude of the centripetal acceleration felt by the drier? 1. A race car is moing with a speed of 200 km/h on a circular section of a race track that has a radius of 400 m. The race car and drier hae a mass of 1400 kg. a) What is the magnitude of the centripetal acceleration felt by the drier? Asked: a c Asked: a c Gien:, r, m Gien:, r, m Relationship: Relationship: 11
1. A race car is moing with a speed of 200 km/h on a circular section of a race track that has a radius of 400 m. The race car and drier hae a mass of 1400 kg. b) What is the centripetal force acting on the mass? 1. A race car is moing with a speed of 200 km/h on a circular section of a race track that has a radius of 400 m. The race car and drier hae a mass of 1400 kg. b) What is the centripetal force acting on the mass? Asked: F c Asked: F c Gien: m Gien: m, a c Relationship: F c Relationship: F c = ma c 2. In the case of a car rounding a turn, what real physical force is acting radially to keep the car moing in a circular path? 2. In the case of a car rounding a turn, what real physical force is acting radially to keep the car moing in a circular path? Static friction between the tires and the road proides the centripetal force. 12