Angle a Block swings while a van turns?

[eatingblaa]

1. A block is hung by a string from the inside roof of a van. When the van goes straight ahead at a speed of 21.5 m/s, the block hangs vertically down. But when the van maintains this same speed around an unbanked curve (radius = 111 m), the block swings toward the outside of the curve. Then the string makes an angle x with the vertical. Find θ.



2. centripetal acceleration Ac = v^2 /r
normal force Fn= mg

i then worked out that the sin x = Ac/Fn

I think these should all be right



3. Ac = 4.164 N

And then sin x = Ac/Fn, but I have no idea where to go from here or even if I am on the right path

 

[tiny-tim]

Hi eatingblaa!

centripetal acceleration Ac = v^2 /r
normal force Fn= mg

i then worked out that the sin x = Ac/Fn

normal to what?

Hint: use Newton's second law (F = ma), in a particular direction.

 

[thomate1]

.

I would first commend you on your initial calculations and for recognizing that the block's movement is due to centripetal acceleration. However, to fully understand the situation, we need to take into account the forces acting on the block and use some basic principles of physics to solve for the angle θ.

First, let's consider the forces acting on the block. As you correctly stated, the block experiences a centripetal acceleration, Ac, towards the center of the circular motion. This is due to the van's velocity and the radius of the curve. The block also experiences a downward force due to gravity, which we can represent as the normal force, Fn, acting in the opposite direction.

Now, let's look at the forces in the vertical direction. Since the block is hanging vertically down when the van is going straight, the forces must be balanced in this direction. This means that the normal force must be equal in magnitude to the weight of the block, mg.

However, when the van turns, the normal force is no longer equal to the weight of the block. The normal force must now provide the necessary centripetal force to keep the block moving in a circular path. This means that the normal force must be greater than the weight of the block, and the difference in these forces is what causes the block to swing towards the outside of the curve.

Now, let's use these principles to solve for the angle θ. We can set up an equation using the forces in the vertical direction:

Fn - mg = 0

Since we know that Fn = mg + Ac, we can substitute this into the equation:

(mg + Ac) - mg = 0

Simplifying, we get:

Ac = mg

Now, we can plug in the values we have for Ac and m (the mass of the block) to solve for g:

g = Ac/m = 4.164 N/ 0.5 kg = 8.328 m/s^2

Finally, we can use this value for g to solve for the angle θ using the relationship sin θ = Ac/Fn:

sin θ = Ac/mg = 4.164 N/ (0.5 kg x 8.328 m/s^2) = 0.998

Taking the inverse sine of both sides, we get:

θ = sin^-1 (0.998) = 89.8 degrees

Therefore,