Malissa+Carvalho

**Third Week **

**1.** An automobile having a mass of 1000 kg is driven into a brick wall in a safety test. The bumper behaves like a spring of constant 5 × 10^6 N/m and compresses 3.16 cm as the car is brought to rest. What was the speed of the car before impact, assuming no energy is lost during impact with the wall?

**Steps: ** >Draw a picture of the problem. >Write down all the givens for the problem (m=1000kg, k=5*10^6 N/m, x=3.16cm). >Convert from cm to m so all the units are the same (x=3.16cm=.0316m). >Determine the formula that needs to be used. Since PE is equal to the KE, energy is conserved. Use the equation (1/2)kx^2=(1/2)mv^2. >Then rearrange the equation for v and solve for v.

 **2.** A block of unknown mass is attached to a spring of spring constant 6 .5 N/m and undergoes simple harmonic motion with an amplitude of 10 cm. When the mass is halfway between its equilibrium position and the endpoint, its speed is measured to be v = 30 cm/s.  i. Calculate the mass of the block.  ii. Find the period of the motion.  iii.Calculate the maximum acceleration of the block.

>Start by drawing a picure of the problem. >Write down the givens for the problem (k=6.5 N/m, x=10cm, v=30cm/s). <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">>Convert the amplitude and the velocity to meters and meters per second (x=10cm=.10m, v=30cm/s=.30m/s) <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">>For part i. the KE is equal to the PE so set the equations equal to each other. (1/2)kx^2=(1/2)mv^2 >Rearange the equation for the mass (m) and solve for m. >For part ii. use the two equations for angular velocity. (w=sqrt(k/m) and w=(2*pi)/T. >Set the equations equal to each other and rearange the equation so T=(2*pi)/(sqrt(k/m)). >Then solve for T. >For part iii. use the two equations for force and set them equal to one another. F=ma F=-kx ma=-kx >Rearange the equation for a and solve for a.
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<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">**3.** A 0.4 kg block attached to a spring of force constant 12 N/m oscillates with an amplitude of 8 cm. <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;"> i. Find the maximum speed of the block. <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;"> ii. Find the speed of the block when it is 4 cm from the equilibrium position. <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;"> iii. Find its acceleration at 4 cm from the equilibrium position.

<span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">>To begin the problem, draw a picture and write down the givens for the problem (m=.4kg, k=12N/m, x=8cm) <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">> Convert from cm to m. (x=8cm=.08m) <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">>In order to fing the maximum speed of the block, you must set the equations of PE and KE equal to each other, since energy is conserved. (1/2)kx^2=(1/2)mv^2 >Rearange the equation for v and slove for v. >For part ii. convert the new x value from cm to m (x=4cm=.04m). >Use the same equations that were used for part i. Set them equal to each other since energy is conserved. >Rearange the equation for v and solve for v. >For part iii. convert the x value from cm to m (x=4cm=.04m). >The equations that are used for this portion of the problem are the force equations. F=ma F=-kx >Set them equal to one another and rearange the equation for a. ma=-kx a=(-kx)/m >Now just solve for a.
 * <span style="font-family: 'Times New Roman',Times,serif; font-size: 110%;">Steps: **