Problem Set #1

Anthony Putrino Physics I Problem Set #1 – Problems 10,12, 22, 50 ,52 10. Acceleration is related to distance and time by the following expression: a=2xt p . Find the power p that makes this equation dimensionally consistent. Answer: Dimensions analysis say that left hand side dimensions is equal to the right hand side dimensions. Dimensions of a is: a=[LT −2 ] Dimensions of x is: x=[L] Dimensions of t is: t=[T] Given data is: a=2xtp Written dimensions on both sides: [LT −2 ]=[L][T]p Compare both sides: p = -2 Final answer is p = -2 12. Newton’s second law states that the acceleration of an object is proportional to the force F acting on it, and inversely proportional to its mass m. That is, a=F/m. What are the dimensions of force? We need to explain the dimensional analysis of mass and acceleration. Explanation: According to Newton's second law states that, acceleration of an object is directly proportional to the force and inversely proportional to its mass. According to Newton's second law, a= F m F=ma Here, a is the acceleration, F is the force, and m is the mass.

The unit for mass is kg and the unit of acceleration is m/s 2 . The dimensional constant of mass is: M= [M 1 L 0 T 0 ] The dimensional constant of acceleration is: a=[M 0 L 1 T -2 ] The dimensional constant of mass is: [M 1 L 0 T 0 ] The dimensional constant of acceleration is: [M 0 L 1 T −2 ] We need to find the dimension of force. Explanation: Dimensional analysis helps us to study the nature of objects. It relates physical quantities of units. The dimensions of force is, F= ma F= [M 1 L 0 T 0 ] × [M 0 L 1 T −2 ] F= [M 1 L 1 T −2 ] Therefore, the dimension of force is F = [M 1 L 1 T −2 ] 22. A car drives on a highway with a speed of 68 mi/h. What is this speed in km/h? V= 68 mi/hr 1 Mile = 1.609 Km 68 Mi = 109.43 Km 68 Mi/Hr = 109.43 km/hr 50. The fastest commercially available rifle bullet claims a muzzle velocity of 4225 ft/s. (a) What is this speed in miles per hour? (b) How far (in meters) can this bullet travel in 5.0 ms? Part A Given the unit of the velocity in the unit of ft/s , v = 4225 ft/s we have the unit conversion relation 1 ft = 0.0001894 miles . ant the time 1 hour = 3600 sec or we can say 1 sec = 1 3600 hour

Explanation: The rate of increase of the newborn baby's brain is given in the question in units of milligram per minute. So, the unit is converted to kilogram per minute (a)Thus, on multiplying R with 24 hours and 60 minutes, the rate of increase in newborn baby's brain mass in one day is obtained. Explanation: One day has 24 hours and 1 hour has 60 minutes. So, to find the rate of increase in mass per day, the rate(R) is multiplied with 24 hour and 60 minute. Thus, the rate of increase of newborn baby's brain, in one day, (R 1 ), is: R 1 =R × 24 hour × 60 minute Substitute, R=1.6×10 −6 kg/minute in the above equation. R 1 =1.6×10 −6 kg/minute × 24 hour × 60 minute R 1 =2.304×10 −3 kg/day Thus the rate of increase of the mass of the brain in one day is, R 1 =2.304×10 −3 kg/day (b)The time(t) required for the mass of the brain to become M, is t= M R 1 Substitute, M=0.0075 kg, and R=2.304×10 −3 kg/day in the above equation, t= 0.0075 kg 2.304 x 10 − 3 kg / day ⇒ t=3.25 days ⇒ t=3.25 days×24 hours×60 minute×60 s ⇒ t=2.801×10 5 s Thus, the time required for the baby's brain's mass to increase by 0.0075 kg is: t=2.801x 10 5 s Final answer: (a) The rate of increase of newborn baby's brain mass in one day is, R 1 =2.304×10 −3 kg/day (b) The time required for the newborn baby's brain mass to increase by 0.0075 kg is 2.801x10 5 s