A solid sphere with a density rho and radius R can be broken up into many small discs of thickness dz as shown, each with a moment of inertia equal to:

 $dI=\frac{1}{2}y^2dm$  
Which can be used to find the moment of inertia of the solid sphere? 

•  $\int_{-R}^R\ \frac{1}{2}\rho\left(R^2-z^2\right)\pi dz$  

•  $\int_{-R}^R\ \frac{1}{2}\rho\left(R^2-y^2\right)^2\pi dy$  

•  $\int_0^R\ \frac{1}{2}\rho\left(R^2-y^2\right)^2\pi dy$  

•  $\int_0^R\ \frac{1}{2}\rho\left(R^2-z^2\right)^2\pi dz$  

The uniform thin rod shown above has mass  and length . What is the moment of the rod about an axis perpendicular to the rod and passing through point , which is halfway between the center and the end of the rod?  
For a rod about its center:

 $I=\frac{1}{12}mL^2$  

•  $\frac{1}{3}ml^2$  

•  $\frac{1}{48}ml^2$  

•  $\frac{7}{48}ml^2$  

•  $\frac{1}{6}ml^2$  

A woman sits on a spinning piano stool with her arms folded. When she extends her arms, which of the following occurs?

• She increases her moment of inertia, thereby increasing her angular speed.

• She increases her moment of inertia, thereby decreasing her angular speed.

• She decreases her moment of inertia, thereby increasing her angular speed.

• She decreases her moment of inertia, thereby decreasing her angular speed.

A ball, rolling without slipping on a horizontal surface, encounters a frictionless, downward sloping ramp, as shown above. Which of the following correctly describes the motion of the ball on the ramp? 

• Increasing translational speed with constant nonzero angular speed 

• Increasing translational speed with decreasing angular speed

• Increasing translational speed with increasing angular speed

• Increasing translational speed with no angular speed 

Tags

NGSS.HS-PS2-1

Two identical cylindrical discs have a common axis. Initially one of the discs is spinning. When the two discs are brought into contact, they stick together. Which of the following is true?

• The total kinetic energy and the total angular momentum are unchanged from their initial values.

• Both the total kinetic energy and the total angular momentum are reduced by half.

• The total angular momentum is unchanged, but the total kinetic energy is reduced by half

• The total angular momentum is reduced by half, but the total kinetic energy is unchanged.

Tags

NGSS.HS-PS2-2

A solid disk of mass M, radius R and rotational inertia I

is free to rotate about a fixed frictionless axis through its center. A force of constant direction and constant magnitude F is exerted on the disk.If the disk accelerates from rest at time t=0, through what angle θ does the disk rotate from t=0 to t=T?

• $\frac{FRT^2}{2M}$

• $\frac{FRT^2}{2I}$

• $\frac{FRT}{I}$

• $\frac{FT^2}{2I}$

Tags

NGSS.HS-PS2-1

A disk initially at rest is acted upon by a constant net torque of τ. The disk spins for a period of t seconds and rotates an angular distance of Δθ. The same disk is brought to rest. The disk is now acted upon by a constant net torque of 2τ. The disk spins for the same period of t seconds. What is the new angular distance the disk rotates?

• $\frac{\Delta\theta}{2}$

• $2\Delta\theta$

• $\sqrt{2}\Delta\theta$

• $4\Delta\theta$

In each case shown below, a 1.0 kg object is released from rest on a ramp of height 2 m from the bottom. Both spheres roll without slipping, and the blocks slide without friction. The ramps are identical in case A and C. The ramps in cases B and D are identical, but not as steep. Rank the speed of the objects when they reach the horizontal surface at the bottom of the ramp.

• B = C > A > D

• D> A > B = C

• C > B > A > D

• B = C > D > A

Tags

NGSS.HS-PS2-1

A turntable with rotational inertia I is spinning on a frictionless axis at an initial angular speed of ω. A funnel is positioned above the turntable and drops syrup in a circle onto the turntable at a radius of R. If the funnel drops the syrup at a rate of S  kg/s, what is the speed of the turntable as a function of time?

• $\omega\left(t\right)=\left(\frac{I}{I+StR}\right)\omega$

• $\omega\left(t\right)=\left(\frac{I}{I+SR^2}\right)\omega$

• $\omega\left(t\right)=\left(\frac{I}{I+StR^2}\right)\omega$

• $\omega\left(t\right)=\left(\frac{2I}{I+Srt}\right)\omega$

Tags

NGSS.HS-PS2-1

NGSS.HS-PS2-2

A disk is initially rotating counterclockwise aroudn a fixed axis with angular speed . At time t = 0, the two forces shown in the figure are exerted on the disk. If counterclockwise is positive, which of the following could show the angular velocity of the disk as a function of time?

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•