Physics 170 - MechanicsLecture 12

Strings & Springs

Pulleys and Ropes An ideal pulley is one that simply changes the

direction of the tension:

Pulleys

Strings and ropes often pass over pulleys that change the direction of the tension. In principle, the friction and inertia in the pulley could modify the transmitted tension. Therefore, it is conventional to assume that such pulleys are massless and frictionless.

Example: Connected Masses

A block of mass m1 is connected by a string and pulley to a hanging mass m2. Find the acceleration a and string tension T.

2,y

1 1

2 2 T2 =T1T2 - m2g = - m2a

Translational Equilibrium

A person lifts a bucket of water from the bottom of a well with a constant speed v. Because the speed is constant, the acceleration must be zero, and the net force on the bucket is zero, so T1 = W.

.

Springs

Stretching a Spring

The unloaded spring has a length L0. Hang a weight of mass m on it and it stretches to a new length L. Δs=L-L0 vs. the applied force Fsp=mg.

We find that Fsp=kΔs, where k is the “spring constant”.

Hooke’s Law for Springs force increases linearly with the amount the

spring is stretched or compressed:

The constant k is called the spring constant.

k has units of N/m or kg/s2.

Hooke’s Law The linear proportionality between force and displacement is found to be valid whether the spring is stretched or compressed, and the force and displacement are always in opposite directions. Therefore, we write the force-displacement relation as:

Spring Forces The force always opposes the compression or extension of the spring.