Physics Problems

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A light string that is attached to a large block of mass 4m passes over a pulley with negligible rotational inertia and is wrapped around a vertical pole of radius r, as shown below. The system is released from rest, and as the block descends the string unwinds and the vertical pole with its attached apparatus rotates. The apparatus consists of a horizontal rod of length 2L, with a small block of mass m attached at each end. The rotational inertia of the pole and the rod are negligible.

A. Determine the rotational inertia of the rod and block apparatus attached at the top of the pole.

B. Determine the downward acceleration of the large block

C. When the large block has descended a distance D, how does the instantaneous total kinetic energy of the three blocks compare with the value 4mgD?
- Greater than 4mgD
- Equal to 4mgD
- Less than 4mgD
The system is now reset. The string is rewound around the pole to bring the large block back to its original location. The small blocks are detached form the rod and then suspended from each end of the rod, using strings of length \(l\) . The system is again released from rest so that as the large block descends and the apparatus rotates, the small blocks swing outward, as shown below.

D. When the large block has descended a distance D, how does the instantaneous total kinetic energy of the three blocks compare to that in part (C), Greater, Equal, or Less?
A uniform disk is mounted to an axle and is free to rotate without friction. A thin uniform rod is rigidly attached to the disk so that it will rotate with the disk. A block is attached to the end of the rod. Properties of the disk, rod, and block are as follows.

Disk: mass = 3m, radius = R, moment of inertia about center \(I_D\) = \(\frac{3}{2}mR^2\)

Rod: mass = m, radius = 2R, moment of inertia about one end \(I_R\) = \(\frac{4}{3}mR^2\)

Block: mass = 2m

The system is held in equilibrium with the rod at an angle \(\theta_0\) to the vertical as shown above, by a horizontal string of negligible mass with one end attached to the disk at the other to a wall. Express your answers to the following in terms of m, R, \(\theta_0\) , and g.

A. Determine the tension in the string

The string is now cut and the disk rod block system is free to rotate

B. Determine the following for the instant immediately after the string is cut
- The magnitude of the angular acceleration of the disk
- The magnitude of the linear acceleration of the mass at teh end of the rod
As the disk rotates the rod passes the horizontal position shown above

C. Determine the linear speed of the mass at the end of the rod for the instant the rod is in the horizontal position.
A solid disk of unknown mass and known radius R is used as a pulley in a lab experiment as shown below. A small block of mass m is attached to a string, the other end of which is attached to the pulley and wrapped around it several times. The block of mass m is released from rest and takes a time t to fall the distance D to the floor.

A. Calculate the linear acceleration \(a\) of the falling block in terms of the given quantities.

B. The time t is measured for various heights D and the data are recorded in the following table.
- What quantities should be graphed in order to determine the acceleration of the block? Explain your reasoning.
- Plot the quantities determined in (B)i, label the axes and draw the best fit line to the data.
- Use your graph to calculate the magnitude of the acceleration.
C. Calculate the rotational inertia of the pulley in terms of m, R, \(a\), and fundamental constants.

D. The value of acceleration found in (B)iii, along with numerical values for the given quantities and your answer to (C), can be used to determine the rotational inertia of the pulley. The pulley is removed from its support and its rotational inertia is found to be greater than this value. Give one explanation for this discrepancy.
A uniform rod of mass M and length L is attached to a pivot of neglible friction as shown below. The pivot is located at a distance \(\frac{L}{3}\) from the left end of the rod. Express all answers in terms of the given quantities and fundamental constants.

A. Calculate the rotational inertia of the rod about the pivot.

B. The rod is then released from rest from teh horizontal positon shown above. Calculate the linear speed of the bottom end of the rod when the rod passes through teh vertical.

C. The rod is brought to rest in teh vertical posoition shown below and hangs freely. It is then displace slightly from this position. Calculate the period of teh osciallation as it swings.
A system consists of a ball of mass \(M_2\) and a uniform rod of mass \(M_1\) and length \(d\). The rod is attached to a horizontal frictionless table by a pivot at point \(P\) and initially rotates at an angular speed \(\omega\), pictured below on the left. The rotational inertia of the rod about point \(P\) is \(\frac{1}{3}M_1d^2\) . The rod strikes the ball, which is initially at rest. As a result of this collision, the rod is stopped and the ball moves in the direction shown on the right. Express all answers in terms of \(M_1\) , \(M_2\) , \(\omega\) , \(d\) , and fundamental constants.

A. Derive an expression for the angular momentum of the rod about point \(P\) before the collision.

B. Derive an expression for the speed \(v\) of the ball after the collision.

C. Assume that this collision is elastic, calculate the numerical value of the ratio \(\frac{M_1}{M_2}\).

D. A new ball with the same mass \(M_1\) as the rod is now placed a distance \(x\) from the pivot, as shown above. Again assuming the collision is elastic, for what value \(x\) will the rod stop moving after hitting the ball?
In March 1999 the Mars Global Surveyor (GS) entered its final orbit about Mars, sending data back to Earth. Assume a circular orbit with a period of 1.18 x 10 \(^2\) minutes = 7.08 x 10 \(^3\) m/s. The mass of the GS is 930 kg and the radius of Mars is 3.43 x 10\(^6\) m .

A. Calculate the radius of the GS orbit.

B. Calculate the mass of Mars.

C. Calculate the total mechanical energy of the GS in this orbit.

D. If the GS was to be placed in a lower circular orbit (closer to the surface of Mars), would the new orbital period of the GS be greater than or less than the given period?

E. In fact, the orbit of the GS entered was slightly eliptical with its closest approach to Mars at 3.71 x 10 \(^5\) m above the surface and its furthest distance at 4.36 x 10 \(^5\) m above the surface. If the speed of the GS at closest approach is 3.40 x 10 \(^3\) m/s, calculate the speed at the furthest point of the orbit.
An oscillator consists of a block of mass 0.500 kg connected to a spring. When set into oscillation with amplitude 35.0 cm, the oscillator repeates its motion every 0.500 s. Find the:

A. period

B. frequency

C. angular frequency

D. spring constant

E. maximum speed

F. magnitude of maximum force on the block from the spring
An oscillator consists of a block attached to a spring( k = 400 N/m). At some time t, the position (measured from the system's equilibrium location), velocity, and acceleration of the block are x = 0.100m, v = -13.6 m/s, and a = -123 m/s\(^2\) . Calculate the

A. frequency of oscillation

B. mass of the block

C. amplitude of the motion
A block is on a horizontal surface ( a shake table ) that is moving back and forth horizontally with simple harmonic motion of frequency 2.0 Hz. The coefficient of static friction between block and surface is 0.50. How great can the amplitude of the simple harmonic motion be if the block is not to slip along the surface?
Two blocks (m = 1.8 kg and M = 10 kg) and a spring ( k = 200 N/m ) are arranged on a horizontal frictionless surface. The coefficient of static friction between the two blocks is 0.40. What amplitude of simple harmonic motion of the spring-block system puts the smaller block on the verge of slipping over the larger block?

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