PHYS 1 Chapter Notes - Chapter 10.1 - 10.3: Normal Force, Kinetic Energy, Angular Acceleration

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6 Apr 2017

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Translational motion = motion of a body as a whole through space. Can be affected by forces acting on the body. Rotational motion depends on magnitude/direction of force, as well as where the force is applied. Torque = the quantitative measure of the tendency of a force to cause/change a body"s rotational motion. Perpendicular distance between the axis of rotation and the line of action of the force. The line along which the force vector lies. Magnitude of torque is the magnitude of the force. Is level arm (perpendicular distance from the line of action to the axis of rotation) Is distance from line of action to axis of rotation. Net torque on a rigid body about an axis causes an angular acceleration about that axis. Only external torques affect a rigid body"s rotation. Angular acceleration of a rotating rigid body is directly proportional to the sum of the torque components along the axis of rotation.

Related Questions

A force F⃗ of magnitude F making an angle θ with the x-axis is applied to a particle located along the axis of rotation A, at Cartesian coordinates (0,0) in the figure. The vector F⃗ lies in the XY plane, and the four axes of rotation A, B, C, and D all lie perpendicular to the XY plane. (Figure 1)

A particle is located at a vector position r⃗ with respect to an axis of rotation (thus r⃗ points from the axis to the point at which the particle is located). The magnitude of the torque τ about this axis due to a force F⃗ acting on the particle is given by

τ=rFsin(α)=(moment arm)⋅F=rF⊥,

where α is the angle between r⃗ and F⃗ , r is the magnitude of r⃗ , F is the magnitude of F⃗ , the component of r⃗ that is perpendicular to F⃗ is the moment arm, and F⊥ is the component of the force that is perpendicular to r⃗ .

Sign convention: You will need to determine the sign by analyzing the direction of the rotation that the torque would tend to produce. Recall that negative torque about an axis corresponds to clockwise rotation.

In this problem, you must express the angle α in the above equation in terms of θ, ϕ, and/or π when entering your answers. Keep in mind that π=180degrees and (π/2)=90 degrees.

Part A

What is the torque τA about axis A due to the force F⃗ ?

Express the torque about axis A at Cartesian coordinates (0,0).

τA =

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Part B

What is the torque τB about axis B due to the force F⃗ ? (B is the point at Cartesian coordinates (0,b), located a distance b from the origin along the y axis.)

Express the torque about axis B in terms of F, θ, ϕ, π, and/or other given coordinate data.

τB =

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Part C

What is the torque τC about axis C due to F⃗ ? (C is the point at Cartesian coordinates (c,0), a distance c along the x axis.)

Express the torque about axis C in terms of F, θ, ϕ, π, and/or other given coordinate data.

τC =

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Part D

What is the torque τD about axis D due to F⃗ ? (D is the point located at a distance d from the origin and making an angle ϕ with the x axis.)

Express the torque about axis D in terms of F, θ, ϕ, π, and/or other given coordinate data.

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The tangential force method involves finding the component of the applied force that is perpendicular to the displacement from the pivot point to where the force is applied. This perpendicular component of the force is called the tangential force.

(a) What is , the magnitude of the tangential force that acts on the pole due to the tension in the rope?

Express your answer in terms of

and

.

When using the tangential force method, you calculate the torque using the equation

$\tau = f_{\rm t} d$ ,

where

is the distance from the pivot to the point where the force is applied. The sign of the torque can be determined by checking which direction the tangential force would tend to cause the pole to rotate (where counterclockwise rotation implies positive torque).

(b) What is the magnitude of the torque

on the pole, about point A, due to the tension in the rope?

Express your answer in terms of

, and

The moment arm method involves finding the effective moment arm of the force. To do this, imagine a line parallel to the force, running through the point at which the force is applied, and extending off to infinity in either direction. You may shift the force vector anywhere you like along this line without changing the torque, provided you do not change the direction of the force vector as you shift it. It is generally most convenient to shift the force vector to a point where the displacement from it to the desired pivot point is perpendicular to its direction. This displacement is called the moment arm.

For example, consider the force due to tension acting on the pole. Shift the force vector to the left, so that it acts at a point directly above point A in the figure. The moment arm of the force is the distance between the pivot and the tail of the shifted force vector. The magnitude of the torque about the pivot is the product of the moment arm and force, and the sign of the torque is again determined by the sense of the rotation of the pole it would cause.

, the length of the moment arm of the force.

Express your answer in terms of

and

(d)Now consider a woman standing on the ball of her foot as shown. A normal force of magnitude

acts upward on the ball of her foot. Achilles' tendon is attached to the back of the foot. The tendon pulls on the small bone in the rear of the foot with a force

. This small bone has a length

, and the angle between this bone and the Achilles' tendon is

. The horizontal displacement between the ball of the foot and the point P is

Find τp, the torque about point P due to the force applied by the Achilles' tendon. Express your answer in terms of F, ϕ, and x.

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Good Vibes: Introduction to Oscillations

The conditions that lead to simple harmonic motion are as follows:

There must be a position of stable equilibrium.

There must be a restoring force acting on the oscillating object. The direction of this force must always point toward the equilibrium, and its magnitude must be directly proportional to the magnitude of the object's displacement from its equilibrium position. Mathematically, the restoring force F⃗ is given by ⃗, where is the displacement from equilibrium and k is a constant that depends on the properties of the oscillating system.

The resistive forces in the system must be reasonably small.

Consider a block of mass m attached to a spring with force constant k, as shown in the figure(Figure 1) . The spring can be either stretched or compressed. The block slides on a frictionless horizontal surface, as shown. When the spring is relaxed, the block is located at x=0. If the block is pulled to the right a distance A and then released, A will be the amplitude of the resulting oscillations.

Assume that the mechanical energy of the block-spring system remains unchanged in the subsequent motion of the block.

Part A

After the block is released from x=A, it will

(a) remain at rest

(b) move to the left until it reaches equilibrium and stop there.

(d) move to the left until it reaches x=−A and then begin to move to the right.

The time it takes the block to complete one cycle is called the period. Usually, the period is denoted T and is measured in seconds.

The frequency, denoted f, is the number of cycles that are completed per unit of time: f=1/T. In SI units, f is measured in inverse seconds, or hertz (Hz).

Part B

If the period is doubled, the frequency is

(a) unchanged

(b) doubled

Part C

An oscillating object takes 0.10 s to complete one cycle; that is, its period is 0.10 s. What is its frequency f?

Express your answer in hertz.

Part D

If the frequency is 40 Hz, what is the period T?

Express your answer in seconds.

The following questions refer to the figure (Figure 2) that graphically depicts the oscillations of the block on the spring.

Note that the vertical axis represents the x-coordinate of the oscillating object, and the horizontal axis represents time.

Part E

Which points on the x-axis are located at a distance A from the equilibrium position?

(a) R only

(b) Q only

Part F

Suppose that the period is T. Which of the following points on the t-axis are separated by the time interval T?

(a) K and L

(b) K and M

(d) L and N

(e) M and P

Now assume for the remaining Parts G - J, that the x coordinate of point R is 0.12 m and the t coordinate of point K is 0.0050 s.

Part G

What is the period T?

Express your answer in seconds.

Part H

How much time t does the block take to travel from the point of maximum displacement to the opposite point of maximum displacement?

Express your answer in seconds.

Part I

What distance d does the object cover during one period of oscillation?

Express your answer in meters.

Part J

What distance d does the object cover between the moments labeled K and N on the graph?

Express your answer in meters.

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PHYS 1 Chapter Notes - Chapter 10.1 - 10.3: Normal Force, Kinetic Energy, Angular Acceleration

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