Energy, Springs, Power, Examples Physics 1D03 - Lecture 22

Example 1 A child of mass 30kg starts on top of a water slide, 6m above the ground. After sliding down to a position of 1m above the ground, the slide curves up and end 1m above the lowest position before the child leaves it and falls into a pool of water. Determine the speed at which the child leaves the slide. Physics 1D03 - Lecture 22

Example 2 A child of mass 30kg starts on top of a water slide, 6m above the ground. After sliding down to a position of 1m above the ground, the slide curves up and end 1m above the lowest position before the child leaves it and falls into a pool of water if the total length of the slide was 20m and there was a constant frictional force of 10 N. Determine the speed at which the child leaves the slide. Physics 1D03 - Lecture 22

Example 3 A mass of 5kg is placed on a vertical spring with a spring constant k=500N/m. What is the maximum compression of the spring ? Physics 1D03 - Lecture 22

Example 4 You slide 20m down a frictionless hill with a slope of 30o starting from rest. At the bottom you collide and stick to another person (at rest) that has 90% of your mass and move on a level frictionless surface. a) Determine the final velocity of the system. b) Determine the velocity if the slope had a coefficient of kinetic friction of 0.1. Physics 1D03 - Lecture 22

Example 5 v0 A block of mass m = 2.0 kg slides at speed v0 = 3.0 m/s along a frictionless table towards a spring of stiffness k = 450 N/m. How far will the spring compress before the block stops? Physics 1D03 - Lecture 22

Example 6 In the figure below, block 2 (mass 1.0 kg) is at rest on a frictionless surface and touching the end of an unstretched spring of spring constant 200 N/m. The other end of the spring is fixed to a wall. Block 1 (mass 2.0 kg) , traveling at speed v1 = 4.0 m/s , collides with block 2, and the two block stick together. When the blocks momentarily stop, by what distance is the spring compressed? Physics 1D03 - Lecture 22

Concept Quiz Two identical vertical springs are compressed by the same amount, one with a heavy ball and one with a light-weight ball. When released, which ball will reach more height? the heavy ball the light ball they will go up the same amount Physics 1D03 - Lecture 22

Power The time rate of doing work is called power. If an external force is applied to an object, and if work is done by this force in a time interval Δt, the average power is defined as: P=W/Δt (unit: J/s = Watt, W) For instantaneous power, we would use the derivative: P=dW/dt And since W=F.s, dW/dt=Fds/dt=F.v, sometimes it is useful to write: P=F . v Physics 1D03 - Lecture 22

Power output A 100W light bulb :100J/s A person can generate about ~ 350 J/s A car engine provides about 110,000 J/s Many common appliances are rated using horsepower (motors for example): 1hp~745 W Mechanical horsepower — 0.745 kW Metric horsepower — 0.735 kW Electrical horsepower — 0.746 kW Boiler horsepower — 9.8095 kW Physics 1D03 - Lecture 22

Example An elevator motor delivers a constant force of 2x105N over a period of 10s as the elevator moves 20m. What is the power ? P=W/t =Fs/t =(2x105N)(20m)/(10s) =4x105 W Physics 1D03 - Lecture 22

10 min rest Physics 1D03 - Lecture 22

Oscillatory Motion - Chapter 14 Kinematics of Simple Harmonic Motion Mass on a spring Energy Knight sections 14.1-14.6 Physics 1D03 - Lecture 22

Oscillatory Motion We have examined the kinematics of linear motion with uniform acceleration. There are other simple types of motion. Many phenomena are repetitive or oscillatory. Example: Block and spring, pendulum, vibrations (musical instruments, molecules) M Physics 1D03 - Lecture 22

Spring and mass M M M Equilibrium: no net force The spring force is always directed back towards equilibrium. This leads to an oscillation of the block about the equilibrium position. M For an ideal spring, the force is proportional to displacement. For this particular force behaviour, the oscillation is simple harmonic motion. x F = -kx Physics 1D03 - Lecture 22

Quiz: You displace a mass from x=0 to x=A and let it go from rest. Where during the motion is acceleration largest? A) at x=0 B) at x=A C) at x=-A D) both at x=A and x=-A Physics 1D03 - Lecture 22

SHM: x(t) A = amplitude t f = phase constant w = angular frequency A is the maximum value of x (x ranges from +A to -A). f gives the initial position at t=0: x(0) = A cosf . w is related to the period T and the frequency f = 1/T T (period) is the time for one complete cycle (seconds). Frequency f (cycles per second or hertz, Hz) is the number of complete cycles per unit time. Physics 1D03 - Lecture 22

The period T of the motion is the time needed to repeat the cycle: The quantity (w t + f) is called the phase, and is measured in radians. The cosine function traces out one complete cycle when the phase changes by 2p radians. The phase is not a physical angle! The period T of the motion is the time needed to repeat the cycle: units: radians/second or s-1 Physics 1D03 - Lecture 22

What is the oscillation period of a FM radio station with a signal at Example - frequency What is the oscillation period of a FM radio station with a signal at 100MHz ? Example - frequency A mass oscillating in SHM starts at x=A and has a period of T. At what time, as a fraction of T, does it first pass through x=A/2? Physics 1D03 - Lecture 22

Velocity and Acceleration a(t) =- w 2 x(t) Physics 1D03 - Lecture 22

Position, Velocity and Acceleration x(t) t v(t) t a(t) t Physics 1D03 - Lecture 22

Question: Where during the motion is the velocity largest? Where during the motion is acceleration largest? When do these happen ? Physics 1D03 - Lecture 22

Example An object oscillates with SHM along the x-axis. Its displacement from the origin varies with time according to the equation: x(t)=(4.0m)cos(πt+π/4) where t is in seconds and the angles in radians. determine the amplitude determine the frequency determine the period its position at t=0 sec calculate the velocity at any time, and the vmax calculate the acceleration at any time, and amax Physics 1D03 - Lecture 22

Example The block is at its equilibrium position and is set in motion by hitting it (and giving it a positive initial velocity vo) at time t = 0. Its motion is SHM with amplitude 5 cm and period 2 seconds. Write the function x(t). M v0 x Result: x(t) = (5 cm) cos[π t – π/2] Physics 1D03 - Lecture 22

10 min rest Physics 1D03 - Lecture 22

When do we have Simple Harmonic Motion ? A system exhibits SHM is we find that acceleration is directly proportional to displacement: a(t) = - w 2 x(t) SHM is also called ‘oscillatory’ motion. Its is called ‘harmonic’ because the sine and cosine function are called harmonic functions, and they are solutions to the above differential equation – lets prove it !!! SHM is ‘periodic’. Physics 1D03 - Lecture 22

Mass and Spring M Newton’s 2nd Law: F = -kx so x This is a 2nd order differential equation for the function x(t). Recall that for SHM, a = -w 2 x : the above is identical except for the proportionality constant. Hence, a spring/mass is a SHO. Hence, we must have: or: Physics 1D03 - Lecture 22

Recall: Velocity and Acceleration We could use x=Asin(ωt+Φ) and obtain the same result Physics 1D03 - Lecture 22

Example A 7.0 kg mass is hung from the bottom end of a vertical spring fastened to the ceiling. The mass is set into vertical oscillations with a period of 2.6 s. Find the spring constant (aka force constant of the spring). Physics 1D03 - Lecture 22

Example A block with a mass of 200g is connected to a light spring with a spring constant k=5.0 N/m and is free to oscillate on a horizontal frictionless surface. The block is displaced 5.0cm from equilibrium and released from rest. find the period of its motion determine the maximum speed of the block determine the maximum acceleration of the block Physics 1D03 - Lecture 22

Example A 1.00 kg mass on a frictionless surface is attached to a horizontal spring. The spring is initially stretched by 0.10 m and the mass is released from rest. The mass moves, and after 0.50 s, the speed of the mass is zero. What is the maximum speed of the mass ??? Physics 1D03 - Lecture 22

Concept Quiz A ball is dropped and keeps bouncing back after hitting the floor. Could this motion be represented by simple harmonic motion equation, x=Asin(ωt+Φ)? Yes No Yes, but only if it bounces to the same height each time Physics 1D03 - Lecture 22