A slender rod is 80.0 cm long and has mass 0.150 kg . A small 0.0200-kg sphere is welded to one end of the rod, and a small 0.0500-kg sphere is welded to the other end. The rod, pivoting about a stationary, frictionless axis at its center, is held horizontal and released from rest.

Explanation:

Given that,

Slender rod

Length of rod=80cm=0.8m

Mass of slender rod=0.12kg

Sphere Bob at one end

Mass M1=0.02kg

Sphere Bod at the other end

Mass M2 =0.05kg

Linear speed of mass 2 at the lowest point

We need to calculate the change in potential of the complete system. m2 and m3 are the masses at the rod ends. note the rod centre of mass neither gains nor loses potential.

So, at the lowest point,

∆U = M2•g•y2 + M1•g•y1

Note, at the lowest point, the mass 1 is 40cm (0.4m) form the midpoint, Also, the mass 2 is -40cm(-04m) from the midpoint

∆U = M2•g•y2 + M1•g•y1

∆U=0.05•9.81•(-0.4) + 0.02•9.82•0.4

∆U=-0.1962+0.07848

∆U=-0.11772 Nm

Now, the moment of inertia of the rod is given as

I=∫r²dm

dm=2pdr

I= 2p∫r²dr

I= 2 × 0.12/0.8 ∫r²dr; from r=0 to 0.4

I=0.3 [r³/3] from r=0 to 0.4

I= 0.3 [ 0.4³/3 -0] ,from r=0 to 0.4

I=0.3 × 0.02133

I=0.0064kg/m².

calculating of inertia of the end masses.

I(1+2)=Σmr² = (m1+m2)r²

I(1+2)=(0.02+0.05)0.4²

I(1+2)=0.07×0.4²

I(1+2)=0.0112 kg/m²

Now, the Energy of the masses due to angular velocity is given as

K.E=½ (I + I(1+2))w²

K.E=½(0.0064+0.0112)w²

K.E= 0.0088w²

Using conservation of energy

The potential energy is equal to the kinetic energy of the system

K.E=P.E

0.0088w²=0.11772

Then, w²=0.11772/0.0088

w²=13.377

w=√13.377

w=3.66rad/s

Then, the relationship between linear velocity and angular velocity is given by

v=wr

v=3.66×0.4

v=1.463m/s

The required linear speed is 1.46m/s approximately

the design of a ski jumping suit involves a trade-off between giving the skiers lift to get higher jumps and ensuring the skier does not face a lot of drag before the jump itself. an expansive suit may to lift the skier higher, but at the same time, it will also increase the air resistance. higher air resistance as the skier goes down the slope limits the speed of the jump. a big suit also increases the risk of losing balance or control, especially if there's a strong gust of wind. while technologically enhanced suits a lot, clothing is not the only factor in a ski jumper’s success. gathering speed while skiing down the slope, ensuring good speed on the ramp, and maintaining good body posture and form in the air all contribute to a ski jumper's success.