Two Wheels and a belt

Two wheels of different radii (large wheel of radius R and small wheel of radius r) rotate by the aid of a belt. The belt move with constant speed. If we assume no friction in the system then:

A) Find the relation between angular speeds of points X and Y.

B) Find the relation between angular speeds of points X and Z.

Analyze the problem:

It is a basic rotational motion problem so we will assume angular speed and its relation with linear speed.

What are the given information?

1. Radius of the large wheel = R
2. Radius of the small wheel = r
3. The belt move with constant speed.. this is the linear speed at the rim of the large and small wheels.

What are the needed information?

1. How angular speed of point x relates to angular speed of point y.
2. How angular speed of point x relates to angular speed of point z.

 

SOLUTION:

Newton’s Laws and Circular Motion

A stone with mass 0.8 kg is attached to the end of 0.9 m string. The string will break if its tension exceeds 60 N. The stone is whirled in a horizontal circle on a frictionless tabletop; the other end of the string remains fixed. (a) draw a free body diagram of the stone. (b) Find the maximum speed the stone can attain without the string breaking.

 

Analyze the problem:

A stone is tied into an end of a string and then swung horizontally to move in circle. As the stone move in circular motion then a radial acceleration is involved. The string can uphold a tension force up to 60 N so there will be a maximum radial acceleration that when exceeded the string will break.

What are the given information?

1. The stone mass, m=0.8 kg
2. The string length, L= 0.9 m
3. Maximum tension, T=60 N

What are the needed information?

1. Body-diagram for the stone.
2. maximum speed of stone so the string is not breaking

SOLUTION:

a)b)

Newton’s Laws Application

Blocks A, B, and C are connected using rope of negligible mass (see figure below). Both A and B weight 23.8 N each, and the coefficient of kinetic friction between each block and the surface is 0.32. Block C descends with constant velocity.

a) Draw free body diagram for blocks A and B.

b) Find tension in rope connecting blocks A and B.

c) What is the weight of block C?

d) If the rope connecting A and B were cut, what would be the acceleration of block C?

Analyze the problem:

The problem has three blocks attached by an inelastic string (so they should have the same acceleration) and pulleys mass ignored. The surface where Blocks A and B are set on have friction and the kinetic friction coefficient was given (kinetic friction coefficient is involved when bodies are in motion which is different than static friction coefficient which used when bodies are not in motion).

What are the given information?

1. Block A weight, FwA=23.8 N
2. Block B weight, FwB=23.8 N
3. Kinetic friction coefficient for all surfaces = 0.32
4. Block C velocity is constant, so acceleration, a =0 m/s2

What are the needed information?

1. Body-diagram draw for blocks A and B.
2. Tension in rope attaching block A and block B.
3. The weight of Block C
4. When rope between block A and block B were cut , then find the acceleration of block C.

SOLUTION:

Note: The acceleration shown in figure below is for part (d).

 

 

Explosion and Collision

A space probe explodes in flight into three equal portions. One portion continues along the original line of flight. The other two go off in directions each inclined at 60º to the original path. The energy released in the explosion is twice as great as the kinetic energy possessed by the probe at the time of the explosion. Determine the kinetic energy of each fragment immediately after the explosion.

Analyze the problem:

With the word explosion we should know that it could involve conservation of momentum and kinetic energy.

What are the given information?

1. Three fragments of exploded probe that have equal masses
2. Direction of probe is φ=0º with the horizontal.
3. Directions of motion of each fragment after explosion are: fragment 1 (θ1=60º with the horizontal), fragment 2 (θ2=0º with the horizontal), fragment 3 (θ3=-60º with the horizontal).
4. Total energy released during explosion is twice the total kinetic energy of the probe.

What are the needed information?

1. The final kinetic energy for each fragment of the probe immediately after explosion.

SOLUTION:

Rotating Flywheel

Delivery trucks which operate by making use of the energy stored in a rotating flywheel have been in use for some time in Germany. The trucks are “charged up” before leaving by using an electric motor to get the flywheel up to its top speed of 6000 revolutions per minute. If one such flywheel is a solid homogeneous cylinder of weight 508 kg and diameter 1.8 m, how long can the truck operate before returning to its base for “recharging”, if its average power requirement is 10 HP?

Analyze the problem:

A disk (flywheel) can rotate with maximum angular speed of 6000 revolutions per minute which can provide kinetic energy that allow a truck to move. When there is an average power needed that allow the truck to operate, the truck will be in motion for limited time as the energy eventually will be reduced with time and that will reduce the power (rate of energy consumed per unit time).

What are the given information?

1. Maximum angular speed of the flywheel, ω=6000 revolutions/minute
2. Flywheel shape is cylinder
3. Flywheel mass, M = 508 kg
4. Flywheel diameter, D= 1.8 m
5. Average power needed for the truck , P = 10 horse power

What are the needed information?

1. The average time the truck can operate

SOLUTION:

Two Blocks on Inclined plane

44. Two blocks are connected by a light string passing over a pulley of radius 0.15 m and moment of inertia I. The blocks move (toward the right) with an acceleration of 1.00 m/s2 along their frictionless inclines (See figure below). (a) Draw free-body diagrams for each of the two blocks and the pulley. (b) Determine FTA and FTB, the tensions in the two parts of the string. (c) Find the net torque acting on the pulley, and determine its moment of inertia, I.

Analyze the problem:

The problem has two blocks on inclined surfaces and they are attached by an inelastic string (so they should have the same acceleration) the pulley’s mass is not ignored (we know that because the problem mentioned that there is a moment of inertia).

What are the given information?

1. Block A mass, mA=8 kg
2. Block B mass, mB=10 kg
3. Acceleration, a = 1 m/s2
4. Inclination Angle 1, θ1=32º
5. Inclination Angle 2, θ2=61º
6. Radius of pulley, R= 0.15 m

What are the needed information?

1. Body-diagram draw for the two blocks and the pulley.
2. FTA and FTB tension forces
3. The net torque on the pulley
4. The moment of inertia of the pulley

SOLUTION:

Atwood Machine

This is a common physics problem and it mentioned in several textbooks.

45. An Atwood machine (shown in figure below) consists of two masses, mA=65 kg and mB=75 kg, connected by a massless inelastic cord that passes over a pulley free to rotate, The pulley is a solid cylinder of radius R=0.45 m and mass 6 kg. (a) Determine the acceleration of each mass. (b) What % error would be made if the moment of inertia of the pulley is ignored? [Hint: the tensions FTA and FTB aare not equal]

Analyze the problem:

It is an Atwood typical problem but here the pulley’s mass  is not ignored and the tensions forces are different for the same cord. Having the pulley’s mass ignored and the tension being the same for the single cord simplify the problem, so here we have to approach the problem in different way but using the same methods (Newton’s laws!).

What are the given information?

1. Block A mass, mA = 65 Kg
2. Block B mass, mB=75 Kg
3. Pulley mass, M=6 kg
4. Pulley Radius, R=0.45 m

What are needed information?

1. Acceleration of both blocks.
2. Percentage error of the two cases (the pulley’s moment of inertia is ignored and the pulley’s moment of inertia is not ignored).

 

SOLUTION:

1. Finding the acceleration of the two blocks:

2. Finding the percentage error:

 

Newton’s laws and weightlessness

6.25 A person stands on a scale in an elevator. As the elevator starts, the scale has a constant reading of 591 N. As the elevator later stops, the scale reading is 391 N. Assume the magnitude of the acceleration is the same during starting and stopping, and determine (a) the weight of the person, (b) the person’s mass, and (c) the acceleration of the elevator.

Analyze the problem:

The problems mentions two stages of motion: (1) starting of motion and (2) stopping of motion for an elevator. The given forces are the normal forces or the readings of the scale.

What are given information:

1. Normal force (N1) when elevator is starting = 591 N
2. Normal force (N2) when elevator is stopping = 391 N

What the problem need:

1. Weight of person
2. Person’s mass
3. Acceleration of the elevator

SOLUTION:

Now, we have to recall Newton’s laws to be able to analyze the motion of man in the elevator.

By the starting of motion an acceleration of the elevator will be directed toward a specific direction and when it stops it will change to the other opposite direction.

Newton’s Laws and Circular Motion

6.11 A 4.00-kg object is attached to a vertical rod by two strings, as in Figure bellow. The object rotates in a horizontal circle at constant speed 6.00 m/s. Find the tension in (a) the upper string and (b) the lower string.

Analyze the problem:

The problems states that it wants tension force in string for the above setting. Newton’s laws should be used beside the dynamic of circular motion concepts.

What are given information:

1. Strings lengths (L) = 2 m
2. Height (h) = 3 m
3. Mass of object (m) = 4 kg
4. Speed of object (v) = 6 m/s

What the problem need:

1. Tension of upper string
2. Tension of lower string

 

SOLUTION:

From above figure we can see that we have isosceles triangle hence we can have right triangle by drawing a perpendicular line on the base of triangle to get sides lengths 2 m, and 3/2=1.5 m, so:

Then:

The tension in the upper string and  the lower string

 

 

 

 

Simple Dynamics of Circular Motion Problem

Let’s analyze the wording of the problem:

1. A conical pendulum is a swinging pendulum that can rotate around an axis. The bob (the mass) will have circular path at all angles θ, but here we need to investigate when the angle is 5°.
2. When a body moves is circular motion it must have at least a radial acceleration (centripetal acceleration).
3. The mass is connected by a wire, the wire will have a tension (because it is used to suspend the bob). The tension should have x- and y-components.

What the information give in problem:

1. The bob’s mass (m) = 80 kg
2. Length of wire (L) = 10 m
3. Angle of pendulum (θ) = 5°

What the problem need:

1. The x- and y-components of tension force.
2. The radial acceleration

 

SOLUTION:

• The x- and y-components of tension force.
  • The x- and y-components of the tension force are:

• The radial acceleration