Question

Please show every step of the solutions. Thanks in advance!

Problem1 . An object of mass m is attached to a vertical rod by two strings of length 2L as shown in the figure on the right. The object rotates in a horizontal circle at a constant speed v. Find the tension in (a) the upper string and (b) the lower string. Problem 2. A race car goes around a circular track. The car increases its speed at a constant rate at as it goes once around the track. Find the angle that the total acceleration of the car makes with the radius connecting the center of the track and the car at the moment the car completes the circle.

Answer 1

Let the tension in the upper string be x and that in the lower string be y.

Balancing vertical forces,

x*1.5/2-y*1.5/2=mg

Balancing horizontal forces,

x*sqrt(2^2-1.5^2)/2+y*sqrt(2^2-1.5^2)/2=mv^2/sqrt(2^2-1.5^2)

Solving for x and y,

x=0.67mg+0.5714mv^2

y=0.5714mv^2-0.67mg

2)Assuming the car begins at rest,

speed at the end of the circle=sqrt(2*a_t*2pi*r)=2*sqrt(a_t*pi*r)

Radial acceleration=v^2/r=(2*a_t*2pi)=4(a_t*pi)

Angle with radius=Tan^-1(a_t*/4(a_t*pi))=Tan^-1(1/4(pi))=4.55 degrees