Unit 06 “Circular Motion, Gravitation and Black Holes” Circular Motion Problem Solving

Centripetal Acceleration Definition Acceleration due to a change in direction. Centripetal acceleration is always directed toward the center of the circle (inward). Equation Vt2 ----------- ac= r ac  centripetal acceleration Vt  tangential speed r  radius

Fc= mVt2 r ----------- Centripetal Force Definition Equation The force that causes circular motion. The centripetal force is always directed toward the center of the circle, inward. Equation mVt2 ----------- Fc= r Fc  centripetal force m  mass Vt  tangential speed r  radius

“Easy” Example 1. A girl swings a bucket of water over her head so that it has a tangential speed of 6m/s. If the rope pulling the bucket is 0.75m long (radius), what is the bucket’s centripetal acceleration?

“Easy” Example ac = Vt2 r ac = __(6m/s)2 0.75m ac = (36m2/s2) 0.75m 1. A girl swings a bucket of water over her head so that it has a tangential speed of 6m/s. If the rope pulling the bucket is 0.75m long (radius), what is the bucket’s centripetal acceleration? ac = Vt2 r ac = Vt = r = ? ac = __(6m/s)2 0.75m 6m/s ac = (36m2/s2) 0.75m 0.75m ac = 48m/s2

Easy 2. What is the centripetal force of a 400kg rollercoaster car being pushed around a track with a radius of 20m if its tangential speed is 15m/s?

Fc = mVt2 r Fc = (400kg)(15m/s)2 20m Fc = (400kg)(225m2/s2) 20m Easy 2. What is the centripetal force of a 400kg rollercoaster car being pushed around a track with a radius of 20m if its tangential speed is 15m/s? Fc = mVt2 r Fc = ? m = 400kg Fc = (400kg)(15m/s)2 20m Vt = 15m/s r = 20m Fc = (400kg)(225m2/s2) 20m Fc = (90000kgm2/s2) 20m Fc = 4500N

Easy + 3. A girl on rollerblades turns a corner with a radius of 8m, if her centripetal acceleration is 10m/s2, what is her tangential speed around the corner?

ac = Vt2 r 10m/s2 =_Vt2_ 8m 80m2/s2 = Vt2 8.94m/s = Vt ac = Vt = r = Easy + 3. A girl on rollerblades turns a corner with a radius of 8m, if her centripetal acceleration is 10m/s2, what is her tangential speed around the corner? ac = Vt2 r ac = Vt = r = 10m/s2 10m/s2 =_Vt2_ 8m x 8m 8m x ? 80m2/s2 = Vt2 8m 8.94m/s = Vt

Easy + 4. A 200 kg bicycle turns a corner with a radius of 15m; a 2500N centripetal force is applied to it by the friction between the ground and the tires, what is the tangential speed of the bicycle?

188m2/s2 = Vt2 Fc = mVt2 r 2500N =_(200kg)Vt2_ 15m Easy + 4. A 200 kg bicycle turns a corner with a radius of 15m; a 2500N centripetal force is applied to it by the friction between the ground and the tires, what is the tangential speed of the bicycle? Fc = mVt2 r 2500N =_(200kg)Vt2_ 15m m = Fc = Vt = r = 200kg 2500N ? 2500N = (13.3kg/m)Vt2 15m 13.3kg/m 13.3kg/m 188m2/s2 = Vt2 13.7m/s = Vt

Medium 5. How long a rope attached to a 20kg bucket if it is pulled inward by a centripetal force of 300N with a tangential speed of 5m/s?

r = 1.67m 300N = (20kg)(5m/s)2 Fc = mVt2 r r 300N = (20kg)(25m2/s2) r Medium 5. How long a rope attached to a 20kg bucket if it is pulled inward by a centripetal force of 300N with a tangential speed of 5m/s? 300N = (20kg)(5m/s)2 r Fc = mVt2 r 300N = (20kg)(25m2/s2) r 300N = _(500kgm2/s2)_ r r x x r r = ? m = 20kg 300N(r) = (500kgm2/s2) Fc = 300N 300N 300N Vt = 5m/s r = 1.67m

Answer the following questions about the horses and the people on the carousel. What is the name of the force causing the horses and people to move in circular motion? In what direction does the force on the horses and people pull? If the carousel suddenly stopped, in which direction would the people move? Why? Which horse is moving faster? The centripetal force Inward Tangent to the circle! Their inertia! They want to move in a straight line! WHAT!? One moves faster!? YES!

A Cartoon Guide to Tangential Speed and “HARD Problems” B: “Sally, would you like to go on a date with me?” S: “Bobby, we will have to see if you are smart enough to be in a relationship with me, where are we going?” B: “I’m going to take you to the Roger Williams Park Carousel!”

S: “A carousel? What is the name of the force that As they walk to Roger Williams Park, Sally begins to ask questions to see how smart Bobby is. S: “A carousel? What is the name of the force that causes the horses and the people to move in a circle on the carousel?” B: “ ___________________________________” S: “In which direction does the force pull on the people and the horses?” B: “__________________________” The centripetal force! Inward!

The floor and the engine S: “What provides the centripetal force?” B: “_______________________________________” S: “If the carousel suddenly stopped and the people weren’t holding on to the horses, which way would the people go? B: “_______________________.” S: “Why?” B: The floor and the engine Tangent! “Because without a force to pull them in a circle, they would move in a straight line.”

S: “You sure are smart Bobby. You might be relationship material.” B: “Great, we are here at the carousel, which horse would you like to go on?” S: “The fastest horse, of course!” B: “The fastest horse?!?!”

Oh no! Which horse is the fastest?!?!

QUICK! I have to calculate which horse moves the fastest, the horse on the inside or the horse on the outside of the carousel.

Inside Horse T = 25s r = 3.5m VT = ? Outside Horse T = 25s r = 5.5m I know… The carousel takes 60seconds to complete one complete circle. The inside horse is 1.5m away from the center of the carousel. The outside horse is 2.9m away from the center of the carousel. Inside Horse T = 25s r = 3.5m VT = ? Outside Horse T = 25s r = 5.5m VT =?

Both horses take the SAME amount of time to complete one revolution. But each horse goes a different distance!     The outside horse has to cover a bigger distance in the same time, greater speed!

Vt = 22.0m/25s Vt = 35.6m/25s Find the speed of the Inside Horse Find the speed of the Outside Horse         s Vt = 22.0m/25s Vt = 35.6m/25s Vt = 0.88m/s Vt = 1.42m/s This makes sense! The _______________horse travels more distance than the ______________horse. They both take the same amount of time to move in a circle. So, the __________________horse must be moving faster! outside inside outside

B: “The faster horse? The faster horse is the horse on the _________________! S: “A person who knows their physics! Perfect! <3” THE END! outside