Suppose we now turn off the motor and apply a brake to bring the grindstone from its greatest angular speed of 50 rad/srad/s to a stop. If the brake generates a constant torque of 30 N⋅mN⋅m , find the total work done (including signs) needed to stop the grindstone. Express your answer in joules.
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An electric motor exerts a constant torque of τ=10N⋅mτ=10N⋅m on a grindstone mounted on its shaft; the moment of inertia of the grindstone is I=2.0kg⋅m2I=2.0kg⋅m2. If the system starts from rest, find the work done by the motor in 8.0 ss and the kinetic energy at the end of this time. What was the average power delivered by the motor? |
SOLUTIONSET UP AND SOLVE We want to use W=τΔθ to find the total work done and then divide that by the time interval Δt=8.0s to find the average power (work per unit time). But first we need to find the total angle Δθ (in radians) through which the grindstone turns. The The angular acceleration is α=τI=10N⋅m2.0kg⋅m2=5.0rad/s2 For constant angular acceleration, θ=θ0+ω0t+12αt2=0+0+12(5.0rad/s2)(8s)2=160radθ=θ0+ω0t+12αt2=0+0+12(5.0rad/s2)(8s)2=160rad The total work is W=τΔθ=(10N⋅m)(160rad)=1600JW=τΔθ=(10N⋅m)(160rad)=1600J Finally, the average power is the total work divided by the time interval: Pav=WΔt=1600J8.0s=200J/s=200WPav=WΔt=1600J8.0s=200J/s=200W REFLECT We can check the energy relationships. The ω=αt=(5.0rad/s2)(8.0s)=40rad/sω=αt=(5.0rad/s2)(8.0s)=40rad/s The grindstone's kinetic energy at this time is K=12Iω2=12(2.0kg⋅m2)(40rad/s)2=1600JK=12Iω2=12(2.0kg⋅m2)(40rad/s)2=1600J This amount equals the total work done, as of course it must. Note that the power is not constant during this process. The torque is constant, but as the angular velocity increases, so does the rate of doing work. The instantaneous power, τωτω, increases linearly with time—from zero, when ω=0ω=0, to 400 WW, with an average value of 200 WW. Part A - Practice Problem:Suppose we now turn off the motor and apply a brake to bring the grindstone from its greatest angular speed of 50 rad/srad/s to a stop. If the brake generates a constant torque of 30 N⋅mN⋅m , find the total work done (including signs) needed to stop the grindstone. Express your answer in joules.
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