Final answer:
The student's question pertains to determining the linear and angular speeds of an object traveling around a circle with a radius of 6 cm, making a central angle of 1/4 radian. Since time is not provided, the speeds are expressed as functions of time: linear speed is 1.5 cm/t and angular speed is (1/4 radian)/t.
Step-by-step explanation:
The question asks for the linear and angular speeds of an object travelling around a circle with a given radius of 6 centimeters and a central angle swept out of 1/4 radian. To find these speeds, we use the relationship between angular speed (ω), linear speed (v), radius (r), and angle (θ):
ω = θ/t and v = r * ω
However, since the question does not provide a time period, we will first determine the arc length swept by the object and relate that to linear speed for an arbitrary time t. The arc length (s) that corresponds to an angle of 1/4 radian on a circle with radius 6 cm is given by s = r * θ. Substituting r = 6 cm and θ = 1/4 radian:
s = 6 cm * 1/4 radian = 1.5 cm
This arc length is the distance the object travels as the angle of 1/4 radian is swept out. If we denote t as the time taken to sweep this angle, then the linear speed (v) of the object is:
v = s/t = 1.5 cm/t
Since ω = θ/t, for θ = 1/4 radian, the angular speed is:
ω = (1/4 radian)/t
Without additional information, we represent the linear and angular speeds as functions of time t. Thus:
Linear Speed (v): 1.5 cm/t
Angular Speed (ω): (1/4 radian)/t