Torque; Physics for PTA

1°

(2 pi/360°) rad = 0.01745 rad

SOLVED PROBLEM 10.2 (a) Express 8° in radians. (b) Express 2.5 rad in degrees. (c) Express π rad in degrees.

a.) theta = 8° (2 pi/360°) = 8 x 0.01745 = 0.14 rad b.) theta =2.5 rad x (360°/2 pi rad) = 2.5 rad x 57.3°/rad = 143 rad c.) theta = pi rad × (360/2 pi rad) = 180°

angular velocity

angular displacement/ time

1 rev/s

2 pi rad/s = 6.28 rad/s

1 rev/min =

2 pi/60 rad/s = 0.105 rad/s

1 revolution (rev)

360° = 2 pi rad

1 rad =

57.3°

period

Any object that travels in uniform circular motion always takes the same amount of time to move completely around the circle

SOLVED PROBLEM 10.11 The starting cord of an outboard motor is wound around a pulley 18 cm in diameterthat is attached to the motor's crankshaft. How much torque is applied to the crankshaftwhen the cord is pulled with a force of 50 N?

Here the moment arm of the force is the pulley's radius of 9 cm = 0.09 m. The torque is therefore τ = Fr = (50 N)(0.09 m) = 4.5 N·m

The torque τ exerted by the force about O has the magnitude

τ = FL Torque = (force)(moment arm)

SOLVED PROBLEM 10.3The shaft of a motor rotates at 1800 rev/min. Through how many radians does it turnin 18 s?

The angular velocity in radians per second that corresponds to 1800 rev/min is w = 1800 rev/min x (2 pi rad/1 rev) x (1 min/60 s) = 189 rad/s Since ω = θ/t, in 18 s the shaft turns through θ = ωt = (189 rad/s)(18 s) = 3402 rad

SOLVED PROBLEM 10.4A phonograph record 30 cm in diameter turns through an angle of 200°. How far doesa point on the rim of the record travel?

The first step is to find the equivalent in radians of 200°, which is theta. So: θ (theta) = 200° (2 pi/360°) = 3.49 rad The radius of the record is 15 cm. Since θ = s/r, a point on its rim travels s = rθ = (15 cm)(3.49 rad) = 52 cm when it turns through 200°.

(Schamns SOLVED PROBLEM 10.19) An air compressor is powered by a 200 rad/s electric motor using a V-belt drive. The motor pulley is 8 cm in radius, and the tension in the V-belt is 135 N on one side and 45 N on the other. Find the power of the motor.

The torque exerted by the motor on the belt is, since the net force on the belt isthe difference between the two tensions, τ = Fr = (135 N − 45 N)(0.08 m) = 7.2 N·m The power output of the motor is thereforeP = τω = (7.2 N·m)(200 rad/s) = 1440 W = 1.44 kW

τ = I α

Torque = (moment of inertia)(angular acceleration)

torque τ (Greek letter tau) =

exerted by a force on a body is a measure of its effectiveness in turning the body about a certain pivot point.

Torque plays the same role in rotational motion that.

force plays in linear motion

the angle θ (Greek letter theta) in radians

is equal to the ratio between the arc s cut by the angle and the radius r of the circle: angle in radians = arc length/radius

In the SI system, the unit of torque

is the newton-meter (N·m);

moment arm of a force F about a pivot point O

is the perpendicular distance L between the line of action of the force and O

torque unit of measurein the British system

it is the pound-foot (lb·ft).

angular velocity

of a body describes how fast it is turning about an axis.

angular velocity is expressed in

rad/s, rev/s or rps, rev/min or rpm

(Schamns SOLVED PROBLEM 10.18) A marine diesel engine develops 90 kW at 2000 rev/min. (a) How much torque can it exert at this velocity? (b) The engine delivers its power through a spur gear 23 cm in radius. If two teeth of the gear transmit torque to another gear at a certain moment, find the force on each gear tooth.

step 1 change 2000 rev/min to rad/s -- 2000 rev/min x 1 min/60 s = 33.33 rev/s. 33.33 rev/s × 2 pi rad/ 1 rev = 209.33 rad/s step 2 find torque with power formula p =Tw; T = p/w = 90000 W/209 rad/s. = 430.6 Nm step 3 find Force from T=Fr F=T/r = 430.6/0.23 m = 1882.26

The torque exerted by a force is also known as

the moment of the force.

(dummies physics) For example, say that you're uniformly spinning a golf ball in a circle at the end of a 1.0-meter string so that it makes one revolution every half-second. How fast is the ball moving? Time to plug in the numbers:

v = 2 π r / T = 2(3.14)(1.0 m) / (0.5 s) = 12.6 m/s