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8.5. PROBLEMS 159

M r₂

r₁ 12. A disk with an inner hub of radius

ri = 0.2m and an outer hub of radius ro = 0.3 m and mass m = 10 kg and moment of inertia Icom = ³₅m r²_o rests on a smooth horizontal surface of unknown coefficients of friction. A string wound around its inner hub is pulled straight downwards by a hanging mass M = 1.0 kg. The disk rolls without slipping. Compute the magnitude and direction of the acceleration of the disk.

Compute the force of friction exerted by the horizontal surface.

T M, R

µ

, µ

13. This spool of mass M = 3.0 kg and radius R = 0.1 m and moment of inertia ³₅M R² is being pulled by a rope of tension T connected to its axle. The coefficients of friction with the ground are µk = 0.2, µs = 0.3. Find the largest T for which the spool rolls without slipping.

14. Suppose that the same spool is pulled by a rope of tension T = 30N connected to its axle. Find Ff, a and α for the spool.

15. What if T = 12N for the spool of problem 13, find a, α and Ff.

16^∗. Find the moment of inertia of these seven disks of mass m (each) and radius R about the axis through the center of the middle disk.

17^∗. Find the moment of inertia of this frame made of four rods of mass m length ℓ about an axis through the geometric center of the square interior.

18^∗. A rod of mass m and length ℓ connected to the ground with a frictionless hinge is released from rest and allowed to fall. Find the speed with which the upper end hits the ground.

M

θ

h

19^∗. This spool of mass M and radius R rolls without slipping to the bottom of the hill, upon reaching it its center of mass speed is measured to be v =

√8Hg

7 . Find its moment of inertia.

N2

θ

T

N1

20. A penguin of mass m_p= 30.0 kg heroically climbs an A-frame ladder (hinged at the top) each side of which has length ℓ = 5.0 m. The angle θ = 45^o. The ladder rests on ice and is stabilized by a wire connecting the centers of the two halves. Find the tension T in the wire, and the forces N1, N2.

v

21. This disk of mass m = 2.0 kg and radius R = 0.1 m compresses a spring of force constant k = 20^N_m by amount x = 0.1 m. When released and disengaged from the spring the disk rolls without slipping at speed v. Find v.

Find its acceleration at the time it is released and begins to move.

v

ω v

22. A sphere of radius R = 0.1 m, mass m2= 2.0 kg and a block of mass m₁ = 3.0 kg compress a spring of force constant k = 200^N_m by amount x = 0.1 m.

When released and disengaged from the spring the sphere rolls without slipping at v₂ and the block slides without friction at v₁. Find both v₁ and v₂.

8.5. PROBLEMS 161

M M M M

m m m

23. Compute the moment of inertia about the origin of this gadget (four masses M at intervals of ℓ on a rod of length 3ℓ mass 3m), and find its kinetic energy when rotated at rate ω. Find the speed of each mass M .

M M M M

m m m

24. Compute the moment of inertia about the origin of this gadget (four masses M at intervals of ℓ on a rod of length 3ℓ mass 3m), and find its kinetic energy when rotated at rate ω. Find the speed of each mass M .

F

25. A stick of length ℓ and mass m ordinarily hangs vertically from its hinge point. What force F must be applied vertically at its center in order to hold it horizontally? What force (magnitude and direction) must the hinge exert?

F

26. A stick of length ℓ and mass m ordinarily hangs vertically from its hinge point. What force F must be applied vertically at its end in order to hold it horizontally? What force (magnitude and direction) must the hinge exert?

27. A stick of length ℓ and mass m ordinarily hangs vertically from its hinge point. What force F must be applied horizontally at its center in order to hold it at an angle θ with respect to vertical? What force (magnitude and direction) must the hinge exert?

28. A stick of length ℓ and mass m ordinarily hangs vertically from its hinge point. What force F must be applied normally at its center in order to hold it at an angle θ with respect to vertical? What force (magnitude and direction) must the hinge exert?

29. For the stick in the previous problem, held in place at rest ant angle θ with respect to vertical; suppose that it is released, find its instantaneous angular acceleration at the moment of release.

30. For the stick in the previous problems, held in place at rest at angle θ with respect to vertical; suppose that it is released, find its speed as it passes through the vertical position.

31. What fraction of a rolling sphere’s kinetic energy is translational? What fraction of a rolling disk’s kinetic energy is translational?

32. A disk of mass m and radius R is given an initial spin rate ω0about its central axis, and is set down on its edge on a table with which it has frictional coefficient µk. Once it begins to roll, find its spin rate and COM speed. How far did it slip before rolling began?

33^∗. In Eq. 8.8 show that unit vectors normal to and tangent to the path of the pendulum bob are, respectively n = cos θ j + sin θi, t =− sin θ j + cos θi

and that

an= a· n = ℓ ˙θ², at= a· t = ℓ¨θ and (this is very useful)

vn = v· n = 0, vt= v· t = ℓ ˙θ

Using the figure for problem 6.32, show that after falling from θ0 = 0 to θ, a pendulum bob of mass m has speed given by

1 2m

( ℓ ˙θ

)2

= mgℓ sin θ and tangential acceleration given by

m (

ℓ¨θ )

= mg cos θ Hint; draw a free-body diagram to get at.

8.5. PROBLEMS 163

θ

a

a

34^∗. A pendulum of mass m length ℓ being at rest in the horizontal position, and is allowed to fall to the point θ illus-trated. Show that the magnitude of its acceleration at that point is

√

a²_n+ a²_t = g

√

1 + 3 sin²θ Hint, use

at= ℓ¨θ = ℓτ /I, an= ℓ ˙θ² and energy conservation.

35^∗. A body of mass m moves on a circle R = 10 m. At a certain time its instantaneous speed is 10 m/s, and the instantaneous rate of change of its speed is 10 m/s². At that instant find the angle between its velocity vector and acceleration vector.

36. A body of mass m moves on a circle R. At a certain time its instantaneous speed is ωR, and the instantaneous rate of change of its speed is αR. At that instant show that the angle between its velocity vector and acceleration vector is tan⁻¹

(ω² α

) .

37. When θ = 45^o in problem 8.34, find the angle between the total acceleration vector and the instantaneous velocity vector of the falling pendulum.

38. Consider a heavy rotating flywheel of mass m = 100 kg and radius R = 0.25 m spinning at ω0= 100 rad/s. A brake applies a force of friction of 100 N to its rim tangentially. Through what total angle θ does it rotate before stopping?

x y

m, L

39. Wind a string of length 5 m around the rim of a wheel of mass m = 1.0 kg and radius R = 0.25 m with its axle held fixed. Pull the string off tangentially with a constant force of 10 N . By time the string has rolled off completely, how fast is the wheel rotating?

40^∗. A equilateral triangular frame is made with three sticks each of mass m = 1.0 kg and length L = 1.0 m.

One corner is at the origin. Find the cent er of mass position (x_com, y_com) and compute the moment of inertial for rotation about the origin.

(F

, F

)

(F

, F

)

ω

ω while in contact with both walls in a corner, with coefficients of friction µk. Find the torque on the disk.

42. A massless level table rests on three equally spaced legs placed on its perimeter, forming an equilateral triangle of side a = 0.5 m. A ball rests on the table. The forces exerted upwards by the legs are 10, 20 and 30 N . What is the mass of the ball, and where is it on the table relative to the leftmost leg (the 10 N leg)?

Chapter 9