Classical mechanics physics, Exercises of Physics

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PHYS 101 MOODLE Exam 3 (02 May 2020, 20:00 – 20 :30) Solution

A pulley of mass 𝑀 and radius 𝑅 (𝐼 =

1

2

2

) is free to rotate around its fixed axis without friction. A spider of mass 𝑚

falls vertically (starting with zero initial velocity) from a point on the ceiling which is a height ℎ above and a distance 𝑅

to the right of the center of the pulley (Figure on the left). The spider holds on to the pulley as soon as it hits, and the

pulley and the spider start rotating. Gravitational acceleration is g.

(a) (5 Pts.) What is the speed of the spider just before it hits the pulley?

During the fall total mechanical energy of the spider is conserved. (2 Pts.) Taking the zero level for gravitational

potential energy as the initial position of the spider, energy conservation means the speed of the spider just before it hits

the pulley is

2 Pts.

𝑓

𝑖

𝑓

2

1 Pt.

𝑓

(b) (5 Pts.) What is the angular speed of the pulley with the spider on it just after the collision?

The axis of the pulley is fixed means there are external forces keeping the axis fixed. Therefore, linear momentum is not

conserved in this collision. However, angular momentum with respect to the axis of the pulley is conserved, because

these forces create no torque with respect to that point. (2 Pts.) The angular speed of the pulley with the spider on it just

after the collision is found as:

2 Pts.

𝑓

𝑖

2

2

1 Pt.

(c) (5 Pts.) Find the angular velocity of the pulley at the time the spider reaches the bottom of the pulley.

Following the collision, total mechanical energy is conserved (2 Pts.). Denoting angular speed at the bottom by 𝜔′,

( 2 Pts. ) 𝐸

𝑓

𝑖

2

2

′

2

2

2

2

ω=?

So

′

2

2

→ ( 1 Pt. ) 𝜔′ =

2

2

2

(d) (5 Pts.) How much energy is lost during the collision?

3 Pts.

𝑓

𝑖

2

2

2

Therefore, the amount energy lost is

( 2 Pts. )